{"id":2200,"date":"2020-08-21T19:42:29","date_gmt":"2020-08-21T23:42:29","guid":{"rendered":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/special-products\/"},"modified":"2025-09-22T19:26:25","modified_gmt":"2025-09-22T23:26:25","slug":"special-products","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/special-products\/","title":{"raw":"8.3 Special Products","rendered":"8.3 Special Products"},"content":{"raw":"[latexpage]\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nBy the end of this section, you will be able to:\n<ul>\n \t<li>Square a binomial using the Binomial Squares Pattern<\/li>\n \t<li>Multiply conjugates using the Product of Conjugates Pattern<\/li>\n \t<li>Recognize and use the appropriate special product pattern<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Square a Binomial Using the Binomial Squares Pattern<\/h1>\n<p id=\"fs-id1169596235990\">Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the <span class=\"no-emphasis\" data-type=\"term\">binomial<\/span> twice and using the methods of the last section, there is less work to do if you learn to use a pattern.<\/p>\n\n<table id=\"eip-940\" style=\"width: 654px; height: 80px;\" summary=\"\/\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">Let's start by looking at \\({\\left(x+9\\right)}^{2}\\).<\/td>\n<td style=\"width: 271.906px; height: 16px;\"><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">What does this mean?<\/td>\n<td style=\"width: 271.906px; height: 16px;\">\\({\\left(x+9\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">It means to multiply \\(\\left(x+9\\right)\\) by itself.<\/td>\n<td style=\"width: 271.906px; height: 16px;\">\\(\\left(x+9\\right)\\left(x+9\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">Then, using FOIL, we get:<\/td>\n<td style=\"width: 271.906px; height: 16px;\">\\({x}^{2}+9x+9x+81\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">Combining like terms gives:<\/td>\n<td style=\"width: 271.906px; height: 16px;\">\\({x}^{2}+18x+81\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-695\" style=\"width: 688px; height: 64px;\" summary=\"\/\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 343.906px; height: 16px;\">Here's another one:<\/td>\n<td style=\"width: 311.906px; height: 16px;\">\\({\\left(y-7\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 343.906px; height: 16px;\">Multiply \\(\\left(y-7\\right)\\) by itself.<\/td>\n<td style=\"width: 311.906px; height: 16px;\">\\(\\left(y-7\\right)\\left(y-7\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 343.906px; height: 16px;\">Using FOIL, we get:<\/td>\n<td style=\"width: 311.906px; height: 16px;\">\\({y}^{2}-7y-7y+49\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 343.906px; height: 16px;\">And combining like terms:<\/td>\n<td style=\"width: 311.906px; height: 16px;\">\\({y}^{2}-14y+49\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-345\" style=\"width: 691px;\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 239.906px;\">And one more:<\/td>\n<td style=\"width: 417.906px;\">\\({\\left(2x+3\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 239.906px;\">Multiply.<\/td>\n<td style=\"width: 417.906px;\">\\(\\left(2x+3\\right)\\left(2x+3\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 239.906px;\">Use FOIL:<\/td>\n<td style=\"width: 417.906px;\">\\(4{x}^{2}+6x+6x+9\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 239.906px;\">Combine like terms.<\/td>\n<td style=\"width: 417.906px;\">\\(4{x}^{2}+12x+9\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169596279334\">Look at these results. Do you see any patterns?<\/p>\nWhat about the number of terms? In each example we squared a binomial and the result was a <span class=\"no-emphasis\" data-type=\"term\">trinomial<\/span>.\n<div id=\"fs-id1169596292349\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({\\left(a+b\\right)}^{2}\\) = ____ + ____ + ____<\/div>\n<p id=\"fs-id1169596275563\">Now look at the <strong data-effect=\"bold\"><em data-effect=\"italics\">first term<\/em><\/strong> in each result. Where did it come from?<\/p>\n<span data-type=\"media\" data-alt=\"This figure has three columns. The first column contains the expression x plus 9, in parentheses, squared. Below this is the product of x plus 9 and x plus 9. Below this is x squared plus 9x plus 9x plus 81. Below this is x squared plus 18x plus 81. The second column contains the expression y minus 7, in parentheses, squared. Below this is the product of y minus 7 and y minus 7. Below this is y squared minus 7y minus 7y plus 49. Below this is the expression y squared minus 14y plus 49. The third column contains the expression 2x plus 3, in parentheses, squared. Below this is the product of 2x plus 3 and 2x plus 3. Below this is 4x squared plus 6x plus 6x plus 9. Below this is 4x squared plus 12x plus 9.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2020\/08\/CNX_ElemAlg_Figure_06_04_001_img_new.jpg\" alt=\"This figure has three columns. The first column contains the expression x plus 9, in parentheses, squared. Below this is the product of x plus 9 and x plus 9. Below this is x squared plus 9x plus 9x plus 81. Below this is x squared plus 18x plus 81. The second column contains the expression y minus 7, in parentheses, squared. Below this is the product of y minus 7 and y minus 7. Below this is y squared minus 7y minus 7y plus 49. Below this is the expression y squared minus 14y plus 49. The third column contains the expression 2x plus 3, in parentheses, squared. Below this is the product of 2x plus 3 and 2x plus 3. Below this is 4x squared plus 6x plus 6x plus 9. Below this is 4x squared plus 12x plus 9.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1169596276671\">The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!<\/p>\n\n<div id=\"fs-id1169596232664\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({\\left(a+b\\right)}^{2}={a}^{2}\\) + ____ + ____<\/div>\n<p id=\"fs-id1169596308108\">To get the <strong data-effect=\"bold\"><em data-effect=\"italics\">first term<\/em><\/strong> of the product, <strong data-effect=\"bold\"><em data-effect=\"italics\">square the first term<\/em><\/strong>.<\/p>\nWhere did the <strong data-effect=\"bold\"><em data-effect=\"italics\">last term<\/em><\/strong> come from? Look at the examples and find the pattern.\n<p id=\"fs-id1169596344338\">The last term is the product of the last terms, which is the square of the last term.<\/p>\n\n<div id=\"fs-id1169596314982\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({\\left(a+b\\right)}^{2}=\\underline{\\qquad}+\\underline{\\qquad}+{b}^{2}\\)<\/div>\n<p id=\"fs-id1169596303982\"><em data-effect=\"italics\">To get the <strong data-effect=\"bold\">last term<\/strong> of the product, <strong data-effect=\"bold\">square the last term<\/strong><\/em>.<\/p>\n<p id=\"fs-id1169596370283\">Finally, look at the <strong data-effect=\"bold\"><em data-effect=\"italics\">middle term<\/em><\/strong>. Notice it came from adding the \u201couter\u201d and the \u201cinner\u201d terms\u2014which are both the same! So the middle term is double the product of the two terms of the binomial.<\/p>\n\n<div id=\"fs-id1169596286712\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({\\left(a+b\\right)}^{2}=\\underline{\\qquad}+ 2ab+ \\underline{\\qquad}\\)<\/div>\n<div class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\({\\left(a-b\\right)}^{2}=\\underline{\\qquad}-2ab + \\underline{\\qquad}\\)<\/div>\n<p id=\"fs-id1169596380513\"><em data-effect=\"italics\">To get the <strong data-effect=\"bold\">middle term<\/strong> of the product, <strong data-effect=\"bold\">multiply the terms and double their product<\/strong><\/em>.<\/p>\n<p id=\"fs-id1169596404678\">Putting it all together:<\/p>\n\n<div id=\"fs-id1169596363994\" data-type=\"note\">\n<div data-type=\"title\">Binomial Squares Pattern<\/div>\n<p id=\"fs-id1169596566195\">If \\(a\\) and \\(b\\) are real numbers,<\/p>\n\\(\\begin{array}{c} {\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\hfill \\\\ {\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}\\hfill \\end{array}\\)\n\n<span id=\"fs-id1168746280275\" data-type=\"media\" data-alt=\"No Alt Text\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_012_img_new.jpg\" alt=\"No Alt Text\" data-media-type=\"image\/jpeg\"><\/span>\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO:<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nTo square a binomial:\n<ul>\n \t<li>square the first term<\/li>\n \t<li>square the last term<\/li>\n \t<li>double their product<\/li>\n<\/ul>\n<\/div>\n<\/div>\nA number example helps verify the pattern.\n\n<\/div>\n<table id=\"eip-234\" class=\"grid\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\({\\left(10+4\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr>\n<td>Square the first term.<\/td>\n<td>\\({10}^{2}+\\underline{\\qquad}+\\)<\/td>\n<\/tr>\n<tr>\n<td>Square the last term.<\/td>\n<td>\\({10}^{2}+\\underline{\\qquad}+{4}^{2}\\)<\/td>\n<\/tr>\n<tr>\n<td>Double their product.<\/td>\n<td>\\({10}^{2}+2\\cdot 10 \\cdot 4+{4}^{2}\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>\\(100+80+16\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>\\(196\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169596567655\">To multiply \\({\\left(10+4\\right)}^{2}\\) usually you\u2019d follow the Order of Operations.<\/p>\n\n<div id=\"fs-id1169596282425\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{c} {\\left(10+4\\right)}^{2} \\\\\u00a0 {\\left(14\\right)}^{2} \\\\ 196 \\end{array}\\)<\/div>\n<p id=\"fs-id1169596497189\">The pattern works!<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596316198\" data-type=\"problem\">\n<p id=\"fs-id1169596285512\">Multiply: \\({\\left(x+5\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596316842\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172183739894\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cSquare the first term.\u201d In the right column is x squared plus blank plus blank. Above the expression is the general form a squared plus 2ab plus b squared. In the third row, the instructions in the left column say \u201cSquare the last term. In the right column is x squared plus blank plus 5 squared. Above this expression is the general form a squared plus 2ab plus b squared. In the fourth row, the instructions in the left column say \u201cDouble their product.\u201d The right column contains the expression x squared plus 2 times x times 5 plus 5squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains x squared plus 10x plus 25.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172183739913\" data-type=\"media\" data-alt=\"x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003a_img_new.jpg\" alt=\"x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the first term.<\/td>\n<td><span id=\"eip-id1172183739930\" data-type=\"media\" data-alt=\"x squared plus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003b_img_new.jpg\" alt=\"x squared plus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the last term.<\/td>\n<td><span id=\"eip-id1172183739947\" data-type=\"media\" data-alt=\"x squared plus blank plus 5 squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003c_img_new.jpg\" alt=\"x squared plus blank plus 5 squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Double the product.<\/td>\n<td><span id=\"eip-id1172183739964\" data-type=\"media\" data-alt=\"x squared plus 2 times x times 5 plus 5 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003d_img_new.jpg\" alt=\"x squared plus 2 times x times 5 plus 5 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172183739981\" data-type=\"media\" data-alt=\"x squared plus 10 x plus 25.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003e_img_new.jpg\" alt=\"x squared plus 10 x plus 25.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596372776\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596299692\" data-type=\"exercise\">\n<div id=\"fs-id1169596299694\" data-type=\"problem\">\n<p id=\"fs-id1169596299697\">Multiply: \\({\\left(x+9\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596306706\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596306708\">\\({x}^{2}+18x+81\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596457234\" data-type=\"problem\">\n<p id=\"fs-id1169596457236\">Multiply: \\({\\left(y+11\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596276678\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596276680\">\\({y}^{2}+22y+121\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596372159\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596372162\" data-type=\"exercise\">\n<div id=\"fs-id1169596276678\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596365345\" data-type=\"problem\">\n<p id=\"fs-id1169596374673\">Multiply: \\({\\left(y-3\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596567926\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172181059928\" class=\"grid\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cSquare the first term.\u201d In the right column is y squared minus blank plus blank. Above the expression is the general form a squared plus 2ab plus b squared. In the third row, the instructions in the left column say \u201cSquare the last term. In the right column is y squared minus blank plus 3 squared. Above this expression is the general form a squared plus 2ab plus b squared. In the fourth row, the instructions in the left column say \u201cDouble their product.\u201d The right column contains the expression y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains y squared minus 6y plus 9.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172181067409\" data-type=\"media\" data-alt=\"y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004a_img_new.jpg\" alt=\"y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the first term.<\/td>\n<td><span id=\"eip-id1172187949388\" data-type=\"media\" data-alt=\"y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004b_img_new.jpg\" alt=\"y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the last term.<\/td>\n<td><span id=\"eip-id1172187949405\" data-type=\"media\" data-alt=\"y squared minus blank plus 3 squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004c_img_new.jpg\" alt=\"y squared minus blank plus 3 squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Double the product.<\/td>\n<td><span id=\"eip-id1172187949422\" data-type=\"media\" data-alt=\"y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004d_img_new.jpg\" alt=\"y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172187949438\" data-type=\"media\" data-alt=\"y squared minus 6 y plus 9.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004e_img_new.jpg\" alt=\"y squared minus 6 y plus 9.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596374747\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596557121\" data-type=\"exercise\">\n<div id=\"fs-id1169596557123\" data-type=\"problem\">\n<p id=\"fs-id1169596557125\">Multiply: \\({\\left(x-9\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596308130\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596308132\">\\({x}^{2}-18x+81\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596378297\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596280554\" data-type=\"exercise\">\n<div id=\"fs-id1169596280556\" data-type=\"problem\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596280556\" data-type=\"problem\">\n<p id=\"fs-id1169596280558\">Multiply: \\({\\left(p-13\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596362582\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596362584\">\\({p}^{2}-26p+169\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596378297\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596280554\" data-type=\"exercise\">\n<div id=\"fs-id1169596362582\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596367138\" data-type=\"problem\">\n<p id=\"fs-id1169596367140\">Multiply: \\({\\left(4x+6\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596384871\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187699262\" class=\"grid\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 4x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cUse the pattern.\u201d In the right column is 4x squared plus 2 times 4x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains 16x squared plus 48x plus 36.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172189408257\" data-type=\"media\" data-alt=\"4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_005a_img_new.jpg\" alt=\"4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172186037824\" data-type=\"media\" data-alt=\"4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_005b_img_new.jpg\" alt=\"4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172188967437\" data-type=\"media\" data-alt=\"16 x squared plus 48 x plus 36.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_005c_img_new.jpg\" alt=\"16 x squared plus 48 x plus 36.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596291797\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596291800\" data-type=\"exercise\">\n<div id=\"fs-id1169596568026\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nMultiply: \\({\\left(6x+3\\right)}^{2}\\).\n\n<details open=\"open\"><summary>Show answer<\/summary>\\(36{x}^{2}+36x+9\\)\n\n<\/details><\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nMultiply: \\({\\left(4x+9\\right)}^{2}\\).\n\n<details open=\"open\"><summary>Show answer<\/summary>\\(16{x}^{2}+72x+81\\)\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596373944\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596454037\" data-type=\"exercise\">\n<div id=\"fs-id1169596450793\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596382663\" data-type=\"problem\">\n<p id=\"fs-id1169596373931\">Multiply: \\({\\left(2x-3y\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596367207\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187673145\" class=\"grid\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 2x minus 3y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cUse the pattern.\u201d In the right column is 2x squared minus 2 times 2x times 3y plus 3y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains 4x squared minus 12xy plus 9y squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172186034852\" data-type=\"media\" data-alt=\"contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_006a_img_new.jpg\" alt=\"contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172186034869\" data-type=\"media\" data-alt=\"2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_006b_img_new.jpg\" alt=\"2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172186034886\" data-type=\"media\" data-alt=\"4 x squared minus 12 x y plus 9 y squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_006c_img_new.jpg\" alt=\"4 x squared minus 12 x y plus 9 y squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596376554\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596376557\" data-type=\"exercise\">\n<div id=\"fs-id1169596302902\" data-type=\"problem\">\n<p id=\"fs-id1169596302904\">Multiply: \\({\\left(2c-d\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596314967\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596314969\">\\(4{c}^{2}-4cd+{d}^{2}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596291727\" data-type=\"problem\">\n<p id=\"fs-id1169596291730\">Multiply: \\({\\left(4x-5y\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596366222\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596366224\">\\(16{x}^{2}-40xy+25{y}^{2}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596371241\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596371244\" data-type=\"exercise\">\n<div id=\"fs-id1169596366222\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596401221\" data-type=\"problem\">\n<p id=\"fs-id1169596401223\">Multiply: \\({\\left(4{u}^{3}+1\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596382867\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172181060733\" class=\"grid\" style=\"height: 86px;\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 4u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cUse the pattern.\u201d In the right column is 4u cubed, in parentheses, squared, plus 2 times 4u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains 16u to the sixth power plus 18u cubed plus 1.\" data-label=\"\">\n<tbody>\n<tr style=\"height: 38px;\">\n<td style=\"height: 38px; width: 239.406px;\"><\/td>\n<td style=\"height: 38px; width: 410.406px;\"><span id=\"eip-id1172181067281\" data-type=\"media\" data-alt=\"4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_007a_img_new.jpg\" alt=\"4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"height: 34px; width: 239.406px;\">Use the pattern.<\/td>\n<td style=\"height: 34px; width: 410.406px;\"><span id=\"eip-id1172181067298\" data-type=\"media\" data-alt=\"4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_007b_img_new.jpg\" alt=\"4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 239.406px;\">Simplify.<\/td>\n<td style=\"height: 14px; width: 410.406px;\"><span id=\"eip-id1172181067314\" data-type=\"media\" data-alt=\"16 u to the sixth power plus 18 u cubed plus 1.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_007c_img_new.jpg\" alt=\"16 u to the sixth power plus 18 u cubed plus 1.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596367883\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596453906\" data-type=\"exercise\">\n<div id=\"fs-id1169596453908\" data-type=\"problem\">\n<p id=\"fs-id1169596453910\">Multiply: \\({\\left(2{x}^{2}+1\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596404465\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596404467\">\\(4{x}^{4}+4{x}^{2}+1\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596378454\" data-type=\"problem\">\n<p id=\"fs-id1169596378457\">Multiply: \\({\\left(3{y}^{3}+2\\right)}^{2}\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596348532\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596284978\">\\(9{y}^{6}+12{y}^{3}+4\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Multiply Conjugates Using the Product of Conjugates Pattern<\/h1>\n<p id=\"fs-id1169596365893\">We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.<\/p>\n<p id=\"fs-id1169596365898\">What do you notice about these pairs of binomials?<\/p>\n\n<div id=\"fs-id1169596570126\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left(x-9\\right)\\left(x+9\\right)\\qquad \\qquad \\left(y-8\\right)\\left(y+8\\right)\\qquad \\qquad\\left(2x-5\\right)\\left(2x+5\\right)\\)<\/div>\n<p id=\"fs-id1169596286685\">Look at the first term of each <span class=\"no-emphasis\" data-type=\"term\">binomial<\/span> in each pair.<\/p>\n<span id=\"fs-id1169596302384\" data-type=\"media\" data-alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_008_img_new.jpg\" alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1169596499807\"><em data-effect=\"italics\">Notice the first terms are the same in each pair.<\/em><\/p>\n<p id=\"fs-id1169596391197\">Look at the last terms of each binomial in each pair.<\/p>\n<span id=\"fs-id1169596391200\" data-type=\"media\" data-alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_009_img_new.jpg\" alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1169596282395\"><em data-effect=\"italics\">Notice the last terms are the same in each pair.<\/em><\/p>\n<p id=\"fs-id1169596285053\"><em data-effect=\"italics\">Notice how each pair has one sum and one difference.<\/em><\/p>\n<span id=\"fs-id1169596405028\" data-type=\"media\" data-alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. Below the x minus 9 is the word \u201cdifference\u201d. Below x plus 9 is the word \u201csum\u201d. The second is y minus 8, in parentheses, times y plus 8, in parentheses. Below y minus 8 is the word \u201cdifference\u201d. Below y plus 8 is the word \u201csum\u201d. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses. Below the 2x minus 5 is the word \u201cdifference\u201d and below 2x plus 5 is the word \u201csum\u201d.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_010_img_new.jpg\" alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. Below the x minus 9 is the word \u201cdifference\u201d. Below x plus 9 is the word \u201csum\u201d. The second is y minus 8, in parentheses, times y plus 8, in parentheses. Below y minus 8 is the word \u201cdifference\u201d. Below y plus 8 is the word \u201csum\u201d. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses. Below the 2x minus 5 is the word \u201cdifference\u201d and below 2x plus 5 is the word \u201csum\u201d.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1169596405024\">A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference has a special name. It is called a <em data-effect=\"italics\">conjugate pair<\/em> and is of the form \\(\\left(a-b\\right),\\left(a+b\\right)\\).<\/p>\n\n<div id=\"fs-id1169596286521\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Conjugate Pair<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169596286526\">A conjugate pair is two binomials of the form<\/p>\n\n<div id=\"fs-id1169596285972\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left(a-b\\right),\\left(a+b\\right)\\).<\/div>\n<\/div>\n<\/div>\nThe pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.\n\n<\/div>\n<\/div>\n<p id=\"fs-id1169596566460\">There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier.<\/p>\n<p id=\"fs-id1169596303706\">Let\u2019s look for the pattern by using FOIL to multiply some conjugate pairs.<\/p>\n\\(\\begin{array}{ccc} \\left(x-9\\right)\\left(x+9\\right) &amp; \\left(y-8\\right)\\left(y+8\\right) &amp; \\left(2x-5\\right)\\left(2x+5\\right) \\\\\u00a0 {x}^{2}+9x-9x-81 \\qquad &amp;{y}^{2}+8y-8y-64 \\qquad &amp; 4{x}^{2}+10x-10x-25 \\\\\u00a0 {x}^{2}-81 &amp; {y}^{2}-64 &amp; 4{x}^{2}-25 \\end{array}\\)\n\n<span id=\"fs-id1169596378050\" data-type=\"media\" data-alt=\"This figure has three columns. The first column contains the product of x plus 9 and x minus 9. Below this is the expression x squared minus 9x plus 9x minus 81. Below this is x squared minus 81. The second column contains the product of y minus 8 and y plus 8. Below this is the expression y squared plus 8y minus 8y minus 64. Below this is y squared minus 64. The third column contains the product of 2x minus 5 and 2x plus 5. Below this is the expression 4x squared plus 10x minus 10x minus 25. Below this is 4x squared minus 25.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_011_img_new.jpg\" alt=\"This figure has three columns. The first column contains the product of x plus 9 and x minus 9. Below this is the expression x squared minus 9x plus 9x minus 81. Below this is x squared minus 81. The second column contains the product of y minus 8 and y plus 8. Below this is the expression y squared plus 8y minus 8y minus 64. Below this is y squared minus 64. The third column contains the product of 2x minus 5 and 2x plus 5. Below this is the expression 4x squared plus 10x minus 10x minus 25. Below this is 4x squared minus 25.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1169596396864\">Each <strong data-effect=\"bold\">first term<\/strong> is the product of the first terms of the binomials, and since they are identical it is the square of the first term.<\/p>\n\\(\\begin{array}{c} \\left(a+b\\right)\\left(a-b\\right)={a}^{2}-\\underline{\\qquad} \\\\\u00a0 \\text{To get the}\\textbf{first term, square the first term}. \\end{array}\\)\n<p id=\"fs-id1169596310955\">The <strong data-effect=\"bold\">last term<\/strong> came from multiplying the last terms, the square of the last term.<\/p>\n\\(\\begin{array}{c} \\left(a+b\\right)\\left(a-b\\right)={a}^{2}-{b}^{2} \\\\ \\text{To get the}\\textbf{last term, square the last term}. \\end{array}\\)\n<p id=\"fs-id1169596378042\">What do you observe about the products?<\/p>\n<p id=\"fs-id1169596378045\">The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.<\/p>\n<p id=\"fs-id1169596303868\">Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.<\/p>\n<p id=\"fs-id1169596389958\">The product of conjugates is always of the form \\({a}^{2}-{b}^{2}\\). This is called a difference of squares.<\/p>\n<p id=\"fs-id1169596391251\">This leads to the pattern:<\/p>\n\n<div id=\"fs-id1169596391023\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Product of Conjugates Pattern<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169596391028\">If \\(a\\) and \\(b\\) are real numbers,<\/p>\n<span id=\"fs-id1169596367551\" data-type=\"media\" data-alt=\"This figure is divided into two sides. On the left side is the following formula: the product of a minus b and a plus b equals a squared minus b squared. On the right side is the same formula labeled: a minus b and a plus b are labeled \u201cconjugates\u201d, the a squared and b squared are labeled squares and the minus sign between the squares is labeled \u201cdifference\u201d. Therefore, the product of two conjugates is called a difference of squares.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_020_img_new.jpg\" alt=\"This figure is divided into two sides. On the left side is the following formula: the product of a minus b and a plus b equals a squared minus b squared. On the right side is the same formula labeled: a minus b and a plus b are labeled \u201cconjugates\u201d, the a squared and b squared are labeled squares and the minus sign between the squares is labeled \u201cdifference\u201d. Therefore, the product of two conjugates is called a difference of squares.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/div>\n<\/div>\nThe product is called a difference of squares.\n\n<\/div>\n<p id=\"fs-id1169596566014\">To multiply conjugates, square the first term, square the last term, and write the product as a difference of squares.<\/p>\n\n<\/div>\n<p id=\"fs-id1169596279072\">Let\u2019s test this pattern with a numerical example.<\/p>\n\n<table id=\"eip-920\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(\\left(10-2\\right)\\left(10+2\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td>It is the product of conjudgates, so the result will be the difference of two squares.<\/td>\n<td>____ \u2013 ____<\/td>\n<\/tr>\n<tr>\n<td>Square the first term.<\/td>\n<td>\\({10}^{2}-\\underline{\\qquad}\\)<\/td>\n<\/tr>\n<tr>\n<td>Square the last term.<\/td>\n<td>\\({10}^{2}-{2}^{2}\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>\\(100-4\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>\\(96\\)<\/td>\n<\/tr>\n<tr>\n<td>What do you get using the order of operations?<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>\\(\\begin{array}{c}\\left(10-2\\right)\\left(10+2\\right)\\\\ \\left(8\\right)\\left(12\\right)\\\\ 96\\end{array}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169596404577\">Notice, the result is the same!<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596404708\" data-type=\"problem\">\n<p id=\"fs-id1169596404710\">Multiply: \\(\\left(x-8\\right)\\left(x+8\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596403359\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1169596403364\">First, recognize this as a product of conjugates. The binomials have the same first terms, and the same last terms, and one binomial is a sum and the other is a difference.<\/p>\n\n<table id=\"eip-id1172188052941\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cIt fits the pattern.\u201d The right column contains the product of x minus 8, in parentheses, and x plus 8, in parentheses. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses. In the second row, the left column says \u201cSquare the first term, x.\u201d The right column contains x squared minus blank. Above this is the general form a squared minus b squared. In the third row, the text on the left says \u201cSquare the last term, 8.\u201d The right column contains the expression x squared minus 8 squared. Above this is the general form a squared minus b squared. In the last row, the text in the left column says \u201cThe product is a difference of squares.\u201d The right column contains the expression x squared minus 64. Above this is the general form a squared minus b squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td>It fits the pattern.<\/td>\n<td><span id=\"eip-id1172188052961\" data-type=\"media\" data-alt=\"The product of x minus 8 and x plus 8. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013a_img_new.jpg\" alt=\"The product of x minus 8 and x plus 8. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the first term, <em data-effect=\"italics\">x<\/em>.<\/td>\n<td><span id=\"eip-id1172188052978\" data-type=\"media\" data-alt=\"x squared minus blank. Above this is the general form a squared minus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013b_img_new.jpg\" alt=\"x squared minus blank. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the last term, 8.<\/td>\n<td><span id=\"eip-id1172188052994\" data-type=\"media\" data-alt=\"x squared minus 8 squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013c_img_new.jpg\" alt=\"x squared minus 8 squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>The product is a difference of squares.<\/td>\n<td><span id=\"eip-id1172188053011\" data-type=\"media\" data-alt=\"x squared minus 64.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013d_img_new.jpg\" alt=\"x squared minus 64.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596382320\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596376846\" data-type=\"exercise\">\n<div id=\"fs-id1169596376848\" data-type=\"problem\">\n\nMultiply: \\(\\left(x-5\\right)\\left(x+5\\right)\\).\n\n<\/div>\n<div id=\"fs-id1169596308784\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596569338\">\\({x}^{2}-25\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596568300\" data-type=\"problem\">\n<p id=\"fs-id1169596568302\">Multiply: \\(\\left(w-3\\right)\\left(w+3\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596569446\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596569448\">\\({w}^{2}-9\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596568294\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596568297\" data-type=\"exercise\">\n<div id=\"fs-id1169596569446\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596361165\" data-type=\"problem\">\n\nMultiply: \\(\\left(2x+5\\right)\\left(2x-5\\right)\\).\n\n<\/div>\n<div id=\"fs-id1169596392529\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1169596566851\">Are the binomials conjugates?<\/p>\n\n<table id=\"eip-id1172187850054\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cIt is the product of conjugates.\u201d The right column contains the product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses. In the second row, the left column says \u201cSquare the first term, 2x.\u201d The right column contains 2x squared minus blank. Above this is the general form a squared minus b squared. In the third row, the text on the left says \u201cSquare the last term, 5.\u201d The right column contains the expression 2x squared minus 5 squared. Above this is the general form a squared minus b squared. In the last row, the text in the left column says \u201cSimplify. The product is a difference of squares.\u201d The right column contains the expression 4x squared minus 25. Above this is the general form a squared minus b squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td>It is the product of conjugates.<\/td>\n<td><span id=\"eip-id1172187850074\" data-type=\"media\" data-alt=\"The product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_014a_img_new.jpg\" alt=\"The product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the first term, 2<em data-effect=\"italics\">x<\/em>.<\/td>\n<td><span id=\"eip-id1172187850091\" data-type=\"media\" data-alt=\"2 x squared minus blank. Above this is the general form a squared minus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_014b_img_new.jpg\" alt=\"2 x squared minus blank. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the last term, 5.<\/td>\n<td><span id=\"eip-id1172187850108\" data-type=\"media\" data-alt=\"2 x squared minus 5 squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_014c_img_new.jpg\" alt=\"2 x squared minus 5 squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify. The product is a difference of squares.<\/td>\n<td><span id=\"eip-id1172187850124\" data-type=\"media\" data-alt=\"4 x squared minus 25.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_014d_img_new.jpg\" alt=\"4 x squared minus 25.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596566475\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596566479\" data-type=\"exercise\">\n<div id=\"fs-id1169596555445\" data-type=\"problem\">\n<p id=\"fs-id1169596555447\">Multiply: \\(\\left(6x+5\\right)\\left(6x-5\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596308819\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596308821\">\\(36{x}^{2}-25\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596366001\" data-type=\"problem\">\n<p id=\"fs-id1169596381369\">Multiply: \\(\\left(2x+7\\right)\\left(2x-7\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596404475\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596404477\">\\(4{x}^{2}-49\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169596276227\">The binomials in the next example may look backwards \u2013 the variable is in the second term. But the two binomials are still conjugates, so we use the same pattern to multiply them.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596276239\" data-type=\"problem\">\n<p id=\"fs-id1169596276241\">Find the product: \\(\\left(3+5x\\right)\\left(3-5x\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596574076\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187955273\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cIt is the product of conjugates.\u201d In the right column is the product of 3 plus 5x and 3 minus 5x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 3 squared minus 5x squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 9 minus 25x squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td>It is the product of conjugates.<\/td>\n<td><span id=\"eip-id1172187955293\" data-type=\"media\" data-alt=\"The product of 3 plus 5 x and 3 minus 5 x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_015a_img_new.jpg\" alt=\"The product of 3 plus 5 x and 3 minus 5 x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172187955310\" data-type=\"media\" data-alt=\"3 squared minus 5 x squared. Above this is the general form a squared minus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_015b_img_new.jpg\" alt=\"3 squared minus 5 x squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172187955326\" data-type=\"media\" data-alt=\"9 minus 25 x squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_015c_img_new.jpg\" alt=\"9 minus 25 x squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596574039\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596574042\" data-type=\"exercise\">\n<div id=\"fs-id1169596574044\" data-type=\"problem\">\n<p id=\"fs-id1169596574046\">Multiply: \\(\\left(7+4x\\right)\\left(7-4x\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596391449\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596391451\">\\(49-16{x}^{2}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596393084\" data-type=\"problem\">\n<p id=\"fs-id1169596393086\">Multiply: \\(\\left(9-2y\\right)\\left(9+2y\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596455633\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596455635\">\\(81-4{y}^{2}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169596455654\">Now we\u2019ll multiply conjugates that have two variables.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596387831\" data-type=\"problem\">\n<p id=\"fs-id1169596387833\">Find the product: \\(\\left(5m-9n\\right)\\left(5m+9n\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596370494\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172182437772\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cThis fits the pattern.\u201d In the right column is the product of 5m minus 9n and 5m plus 9n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 5m squared minus 9n squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 25m squared minus 81n squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td>This fits the pattern.<\/td>\n<td><span id=\"eip-id1172182437792\" data-type=\"media\" data-alt=\"5 m minus 9 n and 5 m plus 9 n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_016a_img_new.jpg\" alt=\"5 m minus 9 n and 5 m plus 9 n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172182437809\" data-type=\"media\" data-alt=\"5 m squared minus 9 n squared. Above this is the general form a squared minus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_016b_img_new.jpg\" alt=\"5 m squared minus 9 n squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172186802474\" data-type=\"media\" data-alt=\"25 m squared minus 81 n squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_016c_img_new.jpg\" alt=\"25 m squared minus 81 n squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596570142\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596570145\" data-type=\"exercise\">\n<div id=\"fs-id1169596570147\" data-type=\"problem\">\n<p id=\"fs-id1169596570149\">Find the product: \\(\\left(4p-7q\\right)\\left(4p+7q\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596364491\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596364494\">\\(16{p}^{2}-49{q}^{2}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596392882\" data-type=\"problem\">\n<p id=\"fs-id1169596392884\">Find the product: \\(\\left(3x-y\\right)\\left(3x+y\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596555078\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596384770\">\\(9{x}^{2}-{y}^{2}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596392876\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596392880\" data-type=\"exercise\">\n<div id=\"fs-id1169596555078\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596360904\" data-type=\"problem\">\n<p id=\"fs-id1169596360906\">Find the product: \\(\\left(cd-8\\right)\\left(cd+8\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596568683\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187818160\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cThis fits the pattern.\u201d In the right column is the product of cd minus 8 and cd plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression cd squared minus 8 squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains c squared d squared minus 64.\" data-label=\"\">\n<tbody>\n<tr>\n<td>This fits the pattern.<\/td>\n<td><span id=\"eip-id1172187818180\" data-type=\"media\" data-alt=\"The product of c d minus 8 and c d plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_017a_img_new.jpg\" alt=\"The product of c d minus 8 and c d plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172188053182\" data-type=\"media\" data-alt=\"c d squared minus 8 squared. Above this is the general form a squared minus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_017b_img_new.jpg\" alt=\"c d squared minus 8 squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172188053198\" data-type=\"media\" data-alt=\"c squared d squared minus 64.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_017c_img_new.jpg\" alt=\"c squared d squared minus 64.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596303932\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596303936\" data-type=\"exercise\">\n<div id=\"fs-id1169596303938\" data-type=\"problem\">\n<p id=\"fs-id1169596303940\">Find the product: \\(\\left(xy-6\\right)\\left(xy+6\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596566590\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596566592\">\\({x}^{2}{y}^{2}-36\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596574406\" data-type=\"problem\">\n<p id=\"fs-id1169596574408\">Find the product: \\(\\left(ab-9\\right)\\left(ab+9\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596366993\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596366995\">\\({a}^{2}{b}^{2}-81\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596574400\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596574404\" data-type=\"exercise\">\n<div id=\"fs-id1169596366993\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596569315\" data-type=\"problem\">\n<p id=\"fs-id1169596569317\">Find the product: \\(\\left(6{u}^{2}-11{v}^{5}\\right)\\left(6{u}^{2}+11{v}^{5}\\right)\\).<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187628858\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cThis fits the pattern.\u201d In the right column is the product of 6u squared minus 11v to the fifth power and 68 squared plus 11v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 6u squared, in parentheses, squared, minus 11v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 36u to the fourth power minus 121v to the tenth power.\" data-label=\"\">\n<tbody>\n<tr>\n<td>This fits the pattern.<\/td>\n<td><span id=\"eip-id1172186035863\" data-type=\"media\" data-alt=\"The product of 6 u squared minus 11 v to the fifth power and 6 u squared plus 11 v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_018a_img_new.jpg\" alt=\"The product of 6 u squared minus 11 v to the fifth power and 6 u squared plus 11 v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172186035880\" data-type=\"media\" data-alt=\"6 u squared, in parentheses, squared, minus 11 v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_018b_img_new.jpg\" alt=\"6 u squared, in parentheses, squared, minus 11 v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172186035897\" data-type=\"media\" data-alt=\"36 u to the fourth power minus 121 v to the tenth power.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_018c_img_new.jpg\" alt=\"36 u to the fourth power minus 121 v to the tenth power.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596302415\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596302419\" data-type=\"exercise\">\n<div id=\"fs-id1169596302421\" data-type=\"problem\">\n<p id=\"fs-id1169596302423\">Find the product: \\(\\left(3{x}^{2}-4{y}^{3}\\right)\\left(3{x}^{2}+4{y}^{3}\\right)\\).<\/p>\n\n<\/div>\n<details><summary>Show answer<\/summary>\n<div id=\"fs-id1169596303989\" data-type=\"solution\"><\/div>\n<\/details>\\(9{x}^{4}-16{y}^{6}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596304022\" data-type=\"problem\">\n<p id=\"fs-id1169596296112\">Find the product: \\(\\left(2{m}^{2}-5{n}^{3}\\right)\\left(2{m}^{2}+5{n}^{3}\\right)\\).<\/p>\n\n<\/div>\n<div id=\"fs-id1169596303217\" data-type=\"solution\"><details><summary>Show answer<\/summary>\n<p id=\"fs-id1169596303219\">\\(4{m}^{4}-25{n}^{6}\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Recognize and Use the Appropriate Special Product Pattern<\/h1>\n<p id=\"fs-id1169596299459\">We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.<\/p>\n\n<div id=\"fs-id1169596299465\" data-type=\"note\">\n<div data-type=\"title\">Comparing the Special Product Patterns<\/div>\n<table id=\"eip-391\" class=\"grid\" summary=\"\/\">\n<tbody>\n<tr>\n<td><strong>Binomial Squares<\/strong><\/td>\n<td><strong>Product of Conjugates<\/strong><\/td>\n<\/tr>\n<tr>\n<td>\\({\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\)<\/td>\n<td>\\(\\left(a-b\\right)\\left(a+b\\right)={a}^{2}-{b}^{2}\\)<\/td>\n<\/tr>\n<tr>\n<td>\\({\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}\\)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>- Squaring a binomial<\/td>\n<td>- Multiplying conjugates<\/td>\n<\/tr>\n<tr>\n<td>- Product is a <strong>trinomial<\/strong><\/td>\n<td>- Product is a <strong>binomial<\/strong><\/td>\n<\/tr>\n<tr>\n<td>- Inner and outer terms with FOIL are <strong>the same.<\/strong><\/td>\n<td>- Inner and outer terms with FOIL are <strong>opposites.<\/strong><\/td>\n<\/tr>\n<tr>\n<td>- Middle term is <strong>double the product<\/strong> of the terms.<\/td>\n<td>- There is <strong>no<\/strong> middle term.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n&nbsp;\n\n<\/div>\n<div id=\"fs-id1169596566225\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596566229\" data-type=\"exercise\">\n<div id=\"fs-id1169596566231\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n\nChoose the appropriate pattern and use it to find the product:\n<p id=\"fs-id1169596499292\">a) \\(\\left(2x-3\\right)\\left(2x+3\\right)\\) b) \\({\\left(5x-8\\right)}^{2}\\) c) \\({\\left(6m+7\\right)}^{2}\\) d) \\(\\left(5x-6\\right)\\left(6x+5\\right)\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169596369746\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<ol id=\"fs-id1168746418593\" class=\"circled\" type=\"a\">\n \t<li>\\(\\left(2x-3\\right)\\left(2x+3\\right)\\) These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.\n<table id=\"eip-id1172186037932\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains the product of 2x minus 3 and 2x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 2x squared minus 3 squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 4x squared minus 9.\" data-label=\"\">\n<tbody>\n<tr>\n<td>This fits the pattern.<\/td>\n<td><span id=\"eip-id1172186037952\" data-type=\"media\" data-alt=\"The product of 2 x minus 3 and 2 x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_019a_img_new.jpg\" alt=\"The product of 2 x minus 3 and 2 x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172185704714\" data-type=\"media\" data-alt=\"2 x squared minus 3 squared. Above this is the general form a squared minus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_019b_img_new.jpg\" alt=\"2 x squared minus 3 squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172185704731\" data-type=\"media\" data-alt=\"4 x squared minus 9.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_019c_img_new.jpg\" alt=\"4 x squared minus 9.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n \t<li>\\({\\left(8x-5\\right)}^{2}\\) We are asked to square a binomial. It fits the <strong data-effect=\"bold\">binomial squares<\/strong> pattern.\n<table id=\"eip-id1172188053327\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 8x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 8x squared minus 2 times 8x times 5 plus 5 squared. Above this is the general form a squared minus 2 times a times b plus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 64x squared minus 80x plus 25.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172184443423\" data-type=\"media\" data-alt=\"8 x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_021a_img_new.jpg\" alt=\"8 x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172184443439\" data-type=\"media\" data-alt=\"8 x squared minus 2 times 8 x times 5 plus 5 squared. Above this is the general form a squared minus 2 a b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_021b_img_new.jpg\" alt=\"8 x squared minus 2 times 8 x times 5 plus 5 squared. Above this is the general form a squared minus 2 a b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172184443456\" data-type=\"media\" data-alt=\"64 x squared minus 80 x plus 25.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_021c_img_new.jpg\" alt=\"64 x squared minus 80 x plus 25.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n \t<li>\\({\\left(6m+7\\right)}^{2}\\) Again, we will square a binomial so we use the <strong data-effect=\"bold\">binomial squares<\/strong> pattern.\n<table id=\"eip-id1172182439091\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 6m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 6m squared plus 2 times 6m times 7 plus 7 squared. Above this is the general form a squared plus 2 times a times b plus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 36m squared plus 84m plus 49.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172187673200\" data-type=\"media\" data-alt=\"6 m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_022a_img_new.jpg\" alt=\"6 m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172187673216\" data-type=\"media\" data-alt=\"6 m squared plus 2 times 6 m times 7 plus 7 squared. Above this is the general form a squared plus 2 a b plus b squared.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_022b_img_new.jpg\" alt=\"6 m squared plus 2 times 6 m times 7 plus 7 squared. Above this is the general form a squared plus 2 a b plus b squared.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172187673233\" data-type=\"media\" data-alt=\"36 m squared plus 84 m plus 49.\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_022c_img_new.jpg\" alt=\"36 m squared plus 84 m plus 49.\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n \t<li>\\(\\left(5x-6\\right)\\left(6x+5\\right)\\) This product does not fit the patterns, so we will use FOIL.\n<table id=\"eip-569\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\\(\\left(5x-6\\right)\\left(6x+5\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td>Use FOIL.<\/td>\n<td>\\(30{x}^{2}+25x-36x-30\\)<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>\\(30{x}^{2}-11x-30\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12.1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596566225\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596566229\" data-type=\"exercise\">\n<div id=\"fs-id1169596566231\" data-type=\"problem\">\n<p id=\"fs-id1169596566233\">Choose the appropriate pattern and use it to find the product:<\/p>\n<p id=\"fs-id1168743981551\">a) \\(\\left(9b-2\\right)\\left(2b+9\\right)\\) b) \\({\\left(9p-4\\right)}^{2}\\) c) \\({\\left(7y+1\\right)}^{2}\\) d) \\(\\left(4r-3\\right)\\left(4r+3\\right)\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169596320033\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596320035\">a) FOIL; \\(18{b}^{2}+77b-18\\) b) Binomial Squares; \\(81{p}^{2}-72p+16\\) c) Binomial Squares; \\(49{y}^{2}+14y+1\\) d) Product of Conjugates; \\(16{r}^{2}-9\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12.2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596567460\" data-type=\"problem\">\n<p id=\"fs-id1169596567463\">Choose the appropriate pattern and use it to find the product:<\/p>\n<p id=\"fs-id1168746404828\">a) \\({\\left(6x+7\\right)}^{2}\\) b) \\(\\left(3x-4\\right)\\left(3x+4\\right)\\) c) \\(\\left(2x-5\\right)\\left(5x-2\\right)\\) d) \\({\\left(6n-1\\right)}^{2}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1169596573993\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\n<p id=\"fs-id1169596573995\">a) Binomial Squares; \\(36{x}^{2}+84x+49\\) b) Product of Conjugates; \\(9{x}^{2}-16\\) c) FOIL; \\(10{x}^{2}-29x+10\\) d) Binomial Squares; \\(36{n}^{2}-12n+1\\)<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\nAccess these online resources for additional instruction and practice with special products:\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596291548\" class=\"media-2\" data-type=\"note\">\n<ul id=\"fs-id1169596291556\" data-display=\"block\">\n \t<li><a href=\"https:\/\/openstax.org\/l\/25Specialprod\">Special Products<\/a><\/li>\n<\/ul>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1169596457039\" data-bullet-style=\"bullet\">\n \t<li><strong data-effect=\"bold\">Binomial Squares Pattern<\/strong>\n<ul id=\"fs-id1169596457051\" data-bullet-style=\"open-circle\">\n \t<li>If \\(a,b\\) are real numbers,\n<span id=\"fs-id1168744116889\" data-type=\"media\" data-alt=\"No Alt Text\"><img src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013_img_new.jpg\" alt=\"No Alt Text\" data-media-type=\"image\/jpeg\"><\/span><\/li>\n \t<li>\\({\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\)<\/li>\n \t<li>\\({\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}\\)<\/li>\n \t<li>To square a binomial: square the first term, square the last term, double their product.<\/li>\n<\/ul>\n<\/li>\n \t<li><strong data-effect=\"bold\">Product of Conjugates Pattern<\/strong>\n<ul data-bullet-style=\"open-circle\">\n \t<li>If \\(a,b\\) are real numbers,\n<span id=\"fs-id1168746511136\" data-type=\"media\" data-alt=\"No Alt Text\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_023_img_new.jpg\" alt=\"No Alt Text\" data-media-type=\"image\/jpeg\"><\/span><\/li>\n \t<li>\\(\\left(a-b\\right)\\left(a+b\\right)={a}^{2}-{b}^{2}\\)<\/li>\n \t<li>The product is called a difference of squares.<\/li>\n<\/ul>\n<\/li>\n \t<li><strong data-effect=\"bold\">To multiply conjugates:<\/strong>\n<ul id=\"fs-id1169596570314\" data-bullet-style=\"open-circle\">\n \t<li><strong data-effect=\"bold\">square the first term square the last term<\/strong> write it as a difference of squares<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1169596373142\">\n \t<dt>conjugate pair<\/dt>\n \t<dd id=\"fs-id1169596568794\">A conjugate pair is two binomials of the form \\(\\left(a-b\\right),\\left(a+b\\right)\\); the pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.<\/dd>\n<\/dl>\n<\/div>\n<h1>Practice Makes Perfect<\/h1>\n<h2 id=\"fs-id1169596570340\">Square a Binomial Using the Binomial Squares Pattern<\/h2>\n<p id=\"fs-id1168746325019\">In the following exercises, square each binomial using the Binomial Squares Pattern.<\/p>\n\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">1. \\({\\left(q+12\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">2. \\({\\left(w+4\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">3. \\({\\left(x+\\dfrac{2}{3}\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">4. \\({\\left(y+\\dfrac{1}{4}\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">5. \\({\\left(y-6\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">6. \\({\\left(b-7\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">7. \\({\\left(p-13\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">8. \\({\\left(m-15\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">9. \\({\\left(4a+10\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">10. \\({\\left(3d+1\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">11. \\({\\left(3z+\\dfrac{1}{5}\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">12. \\({\\left(2q+\\dfrac{1}{3}\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">13. \\({\\left(2y-3z\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">14. \\({\\left(3x-y\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">15. \\({\\left(\\dfrac{1}{8}x-\\dfrac{1}{9}y\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">16. \\({\\left(\\dfrac{1}{5}x-\\dfrac{1}{7}y\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">17. \\({\\left(5{u}^{2}+9\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">18. \\({\\left(3{x}^{2}+2\\right)}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">19. \\({\\left(8{p}^{3}-3\\right)}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">20. \\({\\left(4{y}^{3}-2\\right)}^{2}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span style=\"text-align: initial; background-color: initial;\">In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.<\/span>\n<h2>Multiply Conjugates Using the Product of Conjugates Pattern<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">21. \\(\\left(c-5\\right)\\left(c+5\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">22. \\(\\left(m-7\\right)\\left(m+7\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">23. \\(\\left(b+\\dfrac{6}{7}\\right)\\left(b-\\dfrac{6}{7}\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">24. \\(\\left(x+\\dfrac{3}{4}\\right)\\left(x-\\dfrac{3}{4}\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">25. \\(\\left(8j+4\\right)\\left(8j-4\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">26. \\(\\left(5k+6\\right)\\left(5k-6\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">27. \\(\\left(9c+5\\right)\\left(9c-5\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">28. \\(\\left(11k+4\\right)\\left(11k-4\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">29. \\(\\left(13-q\\right)\\left(13+q\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">30. \\(\\left(11-b\\right)\\left(11+b\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">31. \\(\\left(4-6y\\right)\\left(4+6y\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">32. \\(\\left(5-3x\\right)\\left(5+3x\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">33. \\(\\left(7w+10x\\right)\\left(7w-10x\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">34. \\(\\left(9c-2d\\right)\\left(9c+2d\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">35. \\(\\left(p+\\dfrac{4}{5}q\\right)\\left(p-\\dfrac{4}{5}q\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">36. \\(\\left(m+\\dfrac{2}{3}n\\right)\\left(m-\\dfrac{2}{3}n\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">37. \\(\\left(xy-9\\right)\\left(xy+9\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">38. \\(\\left(ab-4\\right)\\left(ab+4\\right)\\)<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">39. \\(\\left(rs-\\dfrac{2}{7}\\right)\\left(rs+\\dfrac{2}{7}\\right)\\)<\/td>\n<td style=\"width: 50%; height: 14px;\">40. \\(\\left(uv-\\dfrac{3}{5}\\right)\\left(uv+\\dfrac{3}{5}\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">41. \\(\\left(6{m}^{3}-4{n}^{5}\\right)\\left(6{m}^{3}+4{n}^{5}\\right)\\)<\/td>\n<td style=\"width: 50%;\">42. \\(\\left(2{x}^{2}-3{y}^{4}\\right)\\left(2{x}^{2}+3{y}^{4}\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">43. \\(\\left(15{m}^{2}-8{n}^{4}\\right)\\left(15{m}^{2}+8{n}^{4}\\right)\\)<\/td>\n<td style=\"width: 50%;\">44. \\(\\left(12{p}^{3}-11{q}^{2}\\right)\\left(12{p}^{3}+11{q}^{2}\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169596376643\"><strong data-effect=\"bold\">In the following exercises, find each product.<\/strong><\/p>\n\n<h2>Recognize and Use the Appropriate Special Product Pattern<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"fs-id1169596376650\" data-type=\"exercise\">\n<div id=\"fs-id1169596376652\" data-type=\"problem\">\n\n45.\n\na) \\({\\left(2r+12\\right)}^{2}\\)\n\n<\/div>\n<\/div>\n<div id=\"fs-id1169596568003\" data-type=\"exercise\">\n<div id=\"fs-id1169596568005\" data-type=\"problem\">\n\nb) \\(\\left(3p+8\\right)\\left(3p-8\\right)\\)\n\nc) \\(\\left(7a+b\\right)\\left(a-7b\\right)\\)\n\nd) \\({\\left(k-6\\right)}^{2}\\)\n\n<\/div>\n<\/div><\/td>\n<td style=\"width: 50%;\">46.\n\na) \\(\\left(p-3\\right)\\left(p+3\\right)\\)\n\nb) \\({\\left(t-9\\right)}^{2}\\)\n\nc) \\({\\left(m+n\\right)}^{2}\\)\n\nd) \\(\\left(2x+y\\right)\\left(x-2y\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">47.\n\na) \\(\\left({x}^{5}+{y}^{5}\\right)\\left({x}^{5}-{y}^{5}\\right)\\)\n\nb) \\({\\left({m}^{3}-8n\\right)}^{2}\\)\n\nc) \\({\\left(9p+8q\\right)}^{2}\\)\n\nd) \\(\\left({r}^{2}-{s}^{3}\\right)\\left({r}^{3}+{s}^{2}\\right)\\)<\/td>\n<td style=\"width: 50%;\">48.\n\na) \\({\\left({a}^{5}-7b\\right)}^{2}\\)\n\nb) \\(\\left({x}^{2}+8y\\right)\\left(8x-{y}^{2}\\right)\\)\n\nc) \\(\\left({r}^{6}+{s}^{6}\\right)\\left({r}^{6}-{s}^{6}\\right)\\)\n\nd) \\({\\left({y}^{4}+2z\\right)}^{2}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 data-type=\"title\">Everyday Math<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1169596364200\">49. <strong data-effect=\"bold\">Mental math<\/strong> You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as \\(60+5\\).<\/p>\n\n<ol id=\"fs-id1168746270495\" class=\"circled\" type=\"a\">\n \t<li>Multiply \\({\\left(60+5\\right)}^{2}\\) by using the binomial squares pattern, \\({\\left(a+b\\right)}^{2}={a}^{2}+2ab+{b}^{2}\\).<\/li>\n \t<li>Square 65 without using a calculator.<\/li>\n \t<li>Which way is easier for you? Why?<\/li>\n<\/ol>\n<\/td>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1169596568354\">50.<strong data-effect=\"bold\"> Mental math<\/strong> You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as \\(50-3\\) and 53 as \\(50+3\\).<\/p>\n\n<ol id=\"fs-id1168744264659\" class=\"circled\" type=\"a\">\n \t<li>Multiply \\(\\left(50-3\\right)\\left(50+3\\right)\\) by using the product of conjugates pattern, \\(\\left(a-b\\right)\\left(a+b\\right)={a}^{2}-{b}^{2}\\).<\/li>\n \t<li>Multiply \\(47\\cdot 53\\) without using a calculator.<\/li>\n \t<li>Which way is easier for you? Why?<\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 data-type=\"title\">Writing Exercises<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">52. Why does \\({\\left(a+b\\right)}^{2}\\) result in a trinomial, but \\(\\left(a-b\\right)\\left(a+b\\right)\\) result in a binomial?<\/td>\n<td style=\"width: 50%;\">51. How do you decide which pattern to use?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">54. Use the order of operations to show that \\({\\left(3+5\\right)}^{2}\\) is 64, and then use that numerical example to explain why \\({\\left(a+b\\right)}^{2}\\ne {a}^{2}+{b}^{2}\\).<\/td>\n<td style=\"width: 50%;\">\n<div id=\"fs-id1169596387451\" data-type=\"exercise\">\n<div id=\"fs-id1169596387453\" data-type=\"problem\">\n<p id=\"fs-id1169596555462\">53. Marta did the following work on her homework paper:<\/p>\n\\(\\begin{array}{c}{\\left(3-y\\right)}^{2} \\\\{3}^{2}-{y}^{2} \\\\9-{y}^{2} \\end{array}\\)\n<p id=\"fs-id1169596555524\">Explain what is wrong with Marta\u2019s work.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1169596555535\" data-type=\"exercise\">\n<div id=\"fs-id1169596555537\" data-type=\"problem\">\n<p id=\"fs-id1169596555540\"><\/p>\n\n<\/div>\n<\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Answers<\/h1>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">1. \\({q}^{2}+24q+144\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">3. \\({x}^{2}+\\dfrac{4}{3}x+\\dfrac{4}{9}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">5. \\({y}^{2}-12y+36\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">7. \\({p}^{2}-26p+169\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">9. \\(16{a}^{2}+80a+100\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">11. \\(9{z}^{2}+\\dfrac{6}{5}z+\\dfrac{1}{25}\\)<\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 50%; height: 34px;\">13. \\(4{y}^{2}-12yz+9{z}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 34px;\">15. \\(\\dfrac{1}{64}{x}^{2}-\\dfrac{1}{36}xy+\\dfrac{1}{81}{y}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">17. \\(25{u}^{4}+90{u}^{2}+81\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">19. \\(64{p}^{6}-48{p}^{3}+9\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">21. \\({c}^{2}-25\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">23. \\({b}^{2}-\\dfrac{36}{49}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">25. \\(64{j}^{2}-16\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">27. \\(81{c}^{2}-25\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">29. \\(169-{q}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">31. \\(16-36{y}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">33. \\(49{w}^{2}-100{x}^{2}\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">35. \\({p}^{2}-\\dfrac{16}{25}{q}^{2}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">37. \\({x}^{2}{y}^{2}-81\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">39. \\({r}^{2}{s}^{2}-\\dfrac{4}{49}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">41. \\(36{m}^{6}-16{n}^{10}\\)<\/td>\n<td style=\"width: 50%; height: 16px;\">43. \\(225{m}^{4}-64{n}^{8}\\)<\/td>\n<\/tr>\n<tr style=\"height: 70px;\">\n<td style=\"width: 50%; height: 70px;\">45. a) \\(4{r}^{2}+48r+144\\) b) \\(9{p}^{2}-64\\) c) \\(7{a}^{2}-48ab-7{b}^{2}\\) d) \\({k}^{2}-12k+36\\)<\/td>\n<td style=\"width: 50%; height: 70px;\">47. a) \\({x}^{10}-{y}^{10}\\) b) \\({m}^{6}-16{m}^{3}n+64{n}^{2}\\) c) \\(81{p}^{2}+144pq+64{q}^{2}\\) d) \\({r}^{5}+{r}^{2}{s}^{2}-{r}^{3}{s}^{3}-{s}^{5}\\)<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">49. a) 4,225 b) 4,225 c) Answers will vary.<\/td>\n<td style=\"width: 50%; height: 16px;\">51. Answers will vary.<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">53. Answers will vary.<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Attributions<\/h1>\nThis chapter has been adapted from \u201cSpecial Products\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em> (OpenStax)<\/a> by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.\n\n<!-- pb_fixme -->","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Square a binomial using the Binomial Squares Pattern<\/li>\n<li>Multiply conjugates using the Product of Conjugates Pattern<\/li>\n<li>Recognize and use the appropriate special product pattern<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Square a Binomial Using the Binomial Squares Pattern<\/h1>\n<p id=\"fs-id1169596235990\">Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the <span class=\"no-emphasis\" data-type=\"term\">binomial<\/span> twice and using the methods of the last section, there is less work to do if you learn to use a pattern.<\/p>\n<table id=\"eip-940\" style=\"width: 654px; height: 80px;\" summary=\"\/\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">Let&#8217;s start by looking at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fbb904421ab1369a10c2a5bdb9a200dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td style=\"width: 271.906px; height: 16px;\"><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">What does this mean?<\/td>\n<td style=\"width: 271.906px; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fbb904421ab1369a10c2a5bdb9a200dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">It means to multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-67e7eb6cc8c68ea88f7d7b6bfe052a4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -4px;\" \/> by itself.<\/td>\n<td style=\"width: 271.906px; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8f607e6e3041bb189fc595e17232734c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">Then, using FOIL, we get:<\/td>\n<td style=\"width: 271.906px; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d91b3eebb9284b4ac18bf3535302e05f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#43;&#57;&#120;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 383.906px; height: 16px;\">Combining like terms gives:<\/td>\n<td style=\"width: 271.906px; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-312fdde785aba692e6b3c496fe49d1ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-695\" style=\"width: 688px; height: 64px;\" summary=\"\/\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 343.906px; height: 16px;\">Here&#8217;s another one:<\/td>\n<td style=\"width: 311.906px; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2732159116f6f00f83966f1444a4bfef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 343.906px; height: 16px;\">Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c47be1bc77e54e2449a49ccbfda83b87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> by itself.<\/td>\n<td style=\"width: 311.906px; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b6994549f15d4bca2f9b5033b4d88c16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 343.906px; height: 16px;\">Using FOIL, we get:<\/td>\n<td style=\"width: 311.906px; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5ab67866bba6471db03eaa3fd6deecc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#121;&#45;&#55;&#121;&#43;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 343.906px; height: 16px;\">And combining like terms:<\/td>\n<td style=\"width: 311.906px; height: 16px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-cb6137d893dfe84577cfa352de0eca80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#121;&#43;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-345\" style=\"width: 691px;\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 239.906px;\">And one more:<\/td>\n<td style=\"width: 417.906px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c61c1e6b05e1dc0893efd01f132085f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 239.906px;\">Multiply.<\/td>\n<td style=\"width: 417.906px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-91e6b5055aa26c8f49a4272e0206466b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 239.906px;\">Use FOIL:<\/td>\n<td style=\"width: 417.906px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-120a19e3336e4f14fd61e11479a1ff57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#54;&#120;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 239.906px;\">Combine like terms.<\/td>\n<td style=\"width: 417.906px;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a43d3556a76f624112edf1bc919d7cb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169596279334\">Look at these results. Do you see any patterns?<\/p>\n<p>What about the number of terms? In each example we squared a binomial and the result was a <span class=\"no-emphasis\" data-type=\"term\">trinomial<\/span>.<\/p>\n<div id=\"fs-id1169596292349\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fcf4905322cd42d5aefebd64aa6a523b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"58\" style=\"vertical-align: -4px;\" \/> = ____ + ____ + ____<\/div>\n<p id=\"fs-id1169596275563\">Now look at the <strong data-effect=\"bold\"><em data-effect=\"italics\">first term<\/em><\/strong> in each result. Where did it come from?<\/p>\n<p><span data-type=\"media\" data-alt=\"This figure has three columns. The first column contains the expression x plus 9, in parentheses, squared. Below this is the product of x plus 9 and x plus 9. Below this is x squared plus 9x plus 9x plus 81. Below this is x squared plus 18x plus 81. The second column contains the expression y minus 7, in parentheses, squared. Below this is the product of y minus 7 and y minus 7. Below this is y squared minus 7y minus 7y plus 49. Below this is the expression y squared minus 14y plus 49. The third column contains the expression 2x plus 3, in parentheses, squared. Below this is the product of 2x plus 3 and 2x plus 3. Below this is 4x squared plus 6x plus 6x plus 9. Below this is 4x squared plus 12x plus 9.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2020\/08\/CNX_ElemAlg_Figure_06_04_001_img_new.jpg\" alt=\"This figure has three columns. The first column contains the expression x plus 9, in parentheses, squared. Below this is the product of x plus 9 and x plus 9. Below this is x squared plus 9x plus 9x plus 81. Below this is x squared plus 18x plus 81. The second column contains the expression y minus 7, in parentheses, squared. Below this is the product of y minus 7 and y minus 7. Below this is y squared minus 7y minus 7y plus 49. Below this is the expression y squared minus 14y plus 49. The third column contains the expression 2x plus 3, in parentheses, squared. Below this is the product of 2x plus 3 and 2x plus 3. Below this is 4x squared plus 6x plus 6x plus 9. Below this is 4x squared plus 12x plus 9.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169596276671\">The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!<\/p>\n<div id=\"fs-id1169596232664\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-638aaa2b4944e45a11d705be52b99ecd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -5px;\" \/> + ____ + ____<\/div>\n<p id=\"fs-id1169596308108\">To get the <strong data-effect=\"bold\"><em data-effect=\"italics\">first term<\/em><\/strong> of the product, <strong data-effect=\"bold\"><em data-effect=\"italics\">square the first term<\/em><\/strong>.<\/p>\n<p>Where did the <strong data-effect=\"bold\"><em data-effect=\"italics\">last term<\/em><\/strong> come from? Look at the examples and find the pattern.<\/p>\n<p id=\"fs-id1169596344338\">The last term is the product of the last terms, which is the square of the last term.<\/p>\n<div id=\"fs-id1169596314982\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-945344b379dbf49df816b2548ebb4592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;&#43;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"212\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"fs-id1169596303982\"><em data-effect=\"italics\">To get the <strong data-effect=\"bold\">last term<\/strong> of the product, <strong data-effect=\"bold\">square the last term<\/strong><\/em>.<\/p>\n<p id=\"fs-id1169596370283\">Finally, look at the <strong data-effect=\"bold\"><em data-effect=\"italics\">middle term<\/em><\/strong>. Notice it came from adding the \u201couter\u201d and the \u201cinner\u201d terms\u2014which are both the same! So the middle term is double the product of the two terms of the binomial.<\/p>\n<div id=\"fs-id1169596286712\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-37fcdf013781db3a0ee83d95382727f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;&#43;&#32;&#50;&#97;&#98;&#43;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"223\" style=\"vertical-align: -5px;\" \/><\/div>\n<div class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-80ffa17e7a887a4d29eab2a34186d8e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;&#45;&#50;&#97;&#98;&#32;&#43;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"223\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"fs-id1169596380513\"><em data-effect=\"italics\">To get the <strong data-effect=\"bold\">middle term<\/strong> of the product, <strong data-effect=\"bold\">multiply the terms and double their product<\/strong><\/em>.<\/p>\n<p id=\"fs-id1169596404678\">Putting it all together:<\/p>\n<div id=\"fs-id1169596363994\" data-type=\"note\">\n<div data-type=\"title\">Binomial Squares Pattern<\/div>\n<p id=\"fs-id1169596566195\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> are real numbers,<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-81876df9f0b791124d06dc3e676c1b31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"184\" style=\"vertical-align: -17px;\" \/><\/p>\n<p><span id=\"fs-id1168746280275\" data-type=\"media\" data-alt=\"No Alt Text\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_012_img_new.jpg\" alt=\"No Alt Text\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">HOW TO:<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>To square a binomial:<\/p>\n<ul>\n<li>square the first term<\/li>\n<li>square the last term<\/li>\n<li>double their product<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>A number example helps verify the pattern.<\/p>\n<\/div>\n<table id=\"eip-234\" class=\"grid\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9a95133d405b26b6ecc9405301c810a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Square the first term.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8102ad81533ad859ae775599d8a27cab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;&#43;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Square the last term.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0407278462a7fb08ce8252690cd58881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;&#43;&#123;&#52;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Double their product.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4e67686b3487decfd0738ce29868fe75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#48;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#52;&#43;&#123;&#52;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"145\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b1a2bdfef9e1a6dc55808421a74dbb75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#43;&#56;&#48;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"105\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b213e548fae96ae5a5680d5875a42659_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169596567655\">To multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9a95133d405b26b6ecc9405301c810a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/> usually you\u2019d follow the Order of Operations.<\/p>\n<div id=\"fs-id1169596282425\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-462fe3929d37a40fe71b1eac2cac230f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#32;&#92;&#92;&#32;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#32;&#92;&#92;&#32;&#49;&#57;&#54;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"68\" style=\"vertical-align: -23px;\" \/><\/div>\n<p id=\"fs-id1169596497189\">The pattern works!<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596316198\" data-type=\"problem\">\n<p id=\"fs-id1169596285512\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-35e24c74083d85a2eea096d93b19410f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596316842\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172183739894\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cSquare the first term.\u201d In the right column is x squared plus blank plus blank. Above the expression is the general form a squared plus 2ab plus b squared. In the third row, the instructions in the left column say \u201cSquare the last term. In the right column is x squared plus blank plus 5 squared. Above this expression is the general form a squared plus 2ab plus b squared. In the fourth row, the instructions in the left column say \u201cDouble their product.\u201d The right column contains the expression x squared plus 2 times x times 5 plus 5squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains x squared plus 10x plus 25.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172183739913\" data-type=\"media\" data-alt=\"x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003a_img_new.jpg\" alt=\"x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the first term.<\/td>\n<td><span id=\"eip-id1172183739930\" data-type=\"media\" data-alt=\"x squared plus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003b_img_new.jpg\" alt=\"x squared plus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the last term.<\/td>\n<td><span id=\"eip-id1172183739947\" data-type=\"media\" data-alt=\"x squared plus blank plus 5 squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003c_img_new.jpg\" alt=\"x squared plus blank plus 5 squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Double the product.<\/td>\n<td><span id=\"eip-id1172183739964\" data-type=\"media\" data-alt=\"x squared plus 2 times x times 5 plus 5 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003d_img_new.jpg\" alt=\"x squared plus 2 times x times 5 plus 5 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172183739981\" data-type=\"media\" data-alt=\"x squared plus 10 x plus 25.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_003e_img_new.jpg\" alt=\"x squared plus 10 x plus 25.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596372776\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596299692\" data-type=\"exercise\">\n<div id=\"fs-id1169596299694\" data-type=\"problem\">\n<p id=\"fs-id1169596299697\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fbb904421ab1369a10c2a5bdb9a200dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596306706\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596306708\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-312fdde785aba692e6b3c496fe49d1ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596457234\" data-type=\"problem\">\n<p id=\"fs-id1169596457236\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2071674ff9bc6314daf2109d13e55f5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596276678\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596276680\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0dea78244edb73f1bceeea012993105e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#50;&#121;&#43;&#49;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596372159\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596372162\" data-type=\"exercise\">\n<div id=\"fs-id1169596276678\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596365345\" data-type=\"problem\">\n<p id=\"fs-id1169596374673\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f5114b82893660abccb743c511798a2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596567926\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172181059928\" class=\"grid\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cSquare the first term.\u201d In the right column is y squared minus blank plus blank. Above the expression is the general form a squared plus 2ab plus b squared. In the third row, the instructions in the left column say \u201cSquare the last term. In the right column is y squared minus blank plus 3 squared. Above this expression is the general form a squared plus 2ab plus b squared. In the fourth row, the instructions in the left column say \u201cDouble their product.\u201d The right column contains the expression y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains y squared minus 6y plus 9.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172181067409\" data-type=\"media\" data-alt=\"y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004a_img_new.jpg\" alt=\"y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the first term.<\/td>\n<td><span id=\"eip-id1172187949388\" data-type=\"media\" data-alt=\"y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004b_img_new.jpg\" alt=\"y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the last term.<\/td>\n<td><span id=\"eip-id1172187949405\" data-type=\"media\" data-alt=\"y squared minus blank plus 3 squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004c_img_new.jpg\" alt=\"y squared minus blank plus 3 squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Double the product.<\/td>\n<td><span id=\"eip-id1172187949422\" data-type=\"media\" data-alt=\"y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004d_img_new.jpg\" alt=\"y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172187949438\" data-type=\"media\" data-alt=\"y squared minus 6 y plus 9.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_004e_img_new.jpg\" alt=\"y squared minus 6 y plus 9.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596374747\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596557121\" data-type=\"exercise\">\n<div id=\"fs-id1169596557123\" data-type=\"problem\">\n<p id=\"fs-id1169596557125\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-291265d3192a0594146a0b9ba16f2f6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596308130\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596308132\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e8bacd3e6d78eb27a5fa8d8d671b8cea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#120;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596378297\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596280554\" data-type=\"exercise\">\n<div id=\"fs-id1169596280556\" data-type=\"problem\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596280556\" data-type=\"problem\">\n<p id=\"fs-id1169596280558\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-654bbfec32163710abb4a1d570d7edab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596362582\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596362584\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6ab32cd2a6198821a617d180d2ffacf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#54;&#112;&#43;&#49;&#54;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596378297\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596280554\" data-type=\"exercise\">\n<div id=\"fs-id1169596362582\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596367138\" data-type=\"problem\">\n<p id=\"fs-id1169596367140\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6bcf2c7891390c2a58484cf1ad3df6f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596384871\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187699262\" class=\"grid\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 4x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cUse the pattern.\u201d In the right column is 4x squared plus 2 times 4x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains 16x squared plus 48x plus 36.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172189408257\" data-type=\"media\" data-alt=\"4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_005a_img_new.jpg\" alt=\"4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172186037824\" data-type=\"media\" data-alt=\"4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_005b_img_new.jpg\" alt=\"4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172188967437\" data-type=\"media\" data-alt=\"16 x squared plus 48 x plus 36.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_005c_img_new.jpg\" alt=\"16 x squared plus 48 x plus 36.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596291797\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596291800\" data-type=\"exercise\">\n<div id=\"fs-id1169596568026\" data-type=\"problem\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9834c65eced35be96725b66e75603657_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6212e58abbecfb94ddcd335f13073868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#54;&#120;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f6ff655f701aeab12c669321cd779006_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3b425a40fcddbfcf1f50eca738566fb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#50;&#120;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"123\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596373944\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596454037\" data-type=\"exercise\">\n<div id=\"fs-id1169596450793\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596382663\" data-type=\"problem\">\n<p id=\"fs-id1169596373931\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b1906c404a38708134a9d84de9123b12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"79\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596367207\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187673145\" class=\"grid\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 2x minus 3y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cUse the pattern.\u201d In the right column is 2x squared minus 2 times 2x times 3y plus 3y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains 4x squared minus 12xy plus 9y squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172186034852\" data-type=\"media\" data-alt=\"contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_006a_img_new.jpg\" alt=\"contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172186034869\" data-type=\"media\" data-alt=\"2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_006b_img_new.jpg\" alt=\"2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172186034886\" data-type=\"media\" data-alt=\"4 x squared minus 12 x y plus 9 y squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_006c_img_new.jpg\" alt=\"4 x squared minus 12 x y plus 9 y squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596376554\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596376557\" data-type=\"exercise\">\n<div id=\"fs-id1169596302902\" data-type=\"problem\">\n<p id=\"fs-id1169596302904\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f4b7dd0641b915ec72ac0ae0681048ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#99;&#45;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"67\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596314967\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596314969\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d01fe019b6b96095576ed18f982cdcb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#99;&#100;&#43;&#123;&#100;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596291727\" data-type=\"problem\">\n<p id=\"fs-id1169596291730\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f47892031b7baefde63dc1bc68d63ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#53;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"79\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596366222\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596366224\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-be317c1b4ea055bddb3b229703980226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#48;&#120;&#121;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596371241\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596371244\" data-type=\"exercise\">\n<div id=\"fs-id1169596366222\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596401221\" data-type=\"problem\">\n<p id=\"fs-id1169596401223\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-39eda025949b2be34871864963163ca1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596382867\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172181060733\" class=\"grid\" style=\"height: 86px;\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 4u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared. In the second row, the instructions in the left column say \u201cUse the pattern.\u201d In the right column is 4u cubed, in parentheses, squared, plus 2 times 4u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared. In the last row, the left column says \u201cSimplify.\u201d The right column contains 16u to the sixth power plus 18u cubed plus 1.\" data-label=\"\">\n<tbody>\n<tr style=\"height: 38px;\">\n<td style=\"height: 38px; width: 239.406px;\"><\/td>\n<td style=\"height: 38px; width: 410.406px;\"><span id=\"eip-id1172181067281\" data-type=\"media\" data-alt=\"4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_007a_img_new.jpg\" alt=\"4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"height: 34px; width: 239.406px;\">Use the pattern.<\/td>\n<td style=\"height: 34px; width: 410.406px;\"><span id=\"eip-id1172181067298\" data-type=\"media\" data-alt=\"4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_007b_img_new.jpg\" alt=\"4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px; width: 239.406px;\">Simplify.<\/td>\n<td style=\"height: 14px; width: 410.406px;\"><span id=\"eip-id1172181067314\" data-type=\"media\" data-alt=\"16 u to the sixth power plus 18 u cubed plus 1.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_007c_img_new.jpg\" alt=\"16 u to the sixth power plus 18 u cubed plus 1.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596367883\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596453906\" data-type=\"exercise\">\n<div id=\"fs-id1169596453908\" data-type=\"problem\">\n<p id=\"fs-id1169596453910\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-76aa391e41095172487af127e0440b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596404465\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596404467\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d90fd16ff069aedfcde401c945d10161_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#43;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596378454\" data-type=\"problem\">\n<p id=\"fs-id1169596378457\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-718e1261eba12ffdaf2db6cb87f1ab10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596348532\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596284978\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3e486cd03dd4a4ece523f5b64b683da0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#121;&#125;&#94;&#123;&#54;&#125;&#43;&#49;&#50;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Multiply Conjugates Using the Product of Conjugates Pattern<\/h1>\n<p id=\"fs-id1169596365893\">We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.<\/p>\n<p id=\"fs-id1169596365898\">What do you notice about these pairs of binomials?<\/p>\n<div id=\"fs-id1169596570126\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d7dafff58177f90e244b4ec86bc40e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#113;&#113;&#117;&#97;&#100;&#32;&#92;&#113;&#113;&#117;&#97;&#100;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#113;&#113;&#117;&#97;&#100;&#32;&#92;&#113;&#113;&#117;&#97;&#100;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"498\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"fs-id1169596286685\">Look at the first term of each <span class=\"no-emphasis\" data-type=\"term\">binomial<\/span> in each pair.<\/p>\n<p><span id=\"fs-id1169596302384\" data-type=\"media\" data-alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_008_img_new.jpg\" alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169596499807\"><em data-effect=\"italics\">Notice the first terms are the same in each pair.<\/em><\/p>\n<p id=\"fs-id1169596391197\">Look at the last terms of each binomial in each pair.<\/p>\n<p><span id=\"fs-id1169596391200\" data-type=\"media\" data-alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_009_img_new.jpg\" alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. The second is y minus 8, in parentheses, times y plus 8, in parentheses. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169596282395\"><em data-effect=\"italics\">Notice the last terms are the same in each pair.<\/em><\/p>\n<p id=\"fs-id1169596285053\"><em data-effect=\"italics\">Notice how each pair has one sum and one difference.<\/em><\/p>\n<p><span id=\"fs-id1169596405028\" data-type=\"media\" data-alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. Below the x minus 9 is the word \u201cdifference\u201d. Below x plus 9 is the word \u201csum\u201d. The second is y minus 8, in parentheses, times y plus 8, in parentheses. Below y minus 8 is the word \u201cdifference\u201d. Below y plus 8 is the word \u201csum\u201d. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses. Below the 2x minus 5 is the word \u201cdifference\u201d and below 2x plus 5 is the word \u201csum\u201d.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_010_img_new.jpg\" alt=\"This figure has three products. The first is x minus 9, in parentheses, times x plus 9, in parentheses. Below the x minus 9 is the word \u201cdifference\u201d. Below x plus 9 is the word \u201csum\u201d. The second is y minus 8, in parentheses, times y plus 8, in parentheses. Below y minus 8 is the word \u201cdifference\u201d. Below y plus 8 is the word \u201csum\u201d. The last is 2x minus 5, in parentheses, times 2x plus 5, in parentheses. Below the 2x minus 5 is the word \u201cdifference\u201d and below 2x plus 5 is the word \u201csum\u201d.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169596405024\">A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference has a special name. It is called a <em data-effect=\"italics\">conjugate pair<\/em> and is of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-84a71b6db866fdb7531d99a893cd59f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div id=\"fs-id1169596286521\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Conjugate Pair<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169596286526\">A conjugate pair is two binomials of the form<\/p>\n<div id=\"fs-id1169596285972\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-84a71b6db866fdb7531d99a893cd59f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/>.<\/div>\n<\/div>\n<\/div>\n<p>The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1169596566460\">There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier.<\/p>\n<p id=\"fs-id1169596303706\">Let\u2019s look for the pattern by using FOIL to multiply some conjugate pairs.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d0f89d2ce272dcbe8fd9824adf89ed58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#92;&#32;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#120;&#45;&#57;&#120;&#45;&#56;&#49;&#32;&#92;&#113;&#113;&#117;&#97;&#100;&#32;&#38;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#45;&#56;&#121;&#45;&#54;&#52;&#32;&#92;&#113;&#113;&#117;&#97;&#100;&#32;&#38;&#32;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#120;&#45;&#49;&#48;&#120;&#45;&#50;&#53;&#32;&#92;&#92;&#32;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#32;&#38;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;&#32;&#38;&#32;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"542\" style=\"vertical-align: -26px;\" \/><\/p>\n<p><span id=\"fs-id1169596378050\" data-type=\"media\" data-alt=\"This figure has three columns. The first column contains the product of x plus 9 and x minus 9. Below this is the expression x squared minus 9x plus 9x minus 81. Below this is x squared minus 81. The second column contains the product of y minus 8 and y plus 8. Below this is the expression y squared plus 8y minus 8y minus 64. Below this is y squared minus 64. The third column contains the product of 2x minus 5 and 2x plus 5. Below this is the expression 4x squared plus 10x minus 10x minus 25. Below this is 4x squared minus 25.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_011_img_new.jpg\" alt=\"This figure has three columns. The first column contains the product of x plus 9 and x minus 9. Below this is the expression x squared minus 9x plus 9x minus 81. Below this is x squared minus 81. The second column contains the product of y minus 8 and y plus 8. Below this is the expression y squared plus 8y minus 8y minus 64. Below this is y squared minus 64. The third column contains the product of 2x minus 5 and 2x plus 5. Below this is the expression 4x squared plus 10x minus 10x minus 25. Below this is 4x squared minus 25.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1169596396864\">Each <strong data-effect=\"bold\">first term<\/strong> is the product of the first terms of the binomials, and since they are identical it is the square of the first term.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-98c342c87a39915e7182622286ddaffe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;&#32;&#92;&#92;&#32;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#103;&#101;&#116;&#32;&#116;&#104;&#101;&#125;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#44;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#125;&#46;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"367\" style=\"vertical-align: -15px;\" \/><\/p>\n<p id=\"fs-id1169596310955\">The <strong data-effect=\"bold\">last term<\/strong> came from multiplying the last terms, the square of the last term.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f1083ae2dd3bd299d72d58c723a1733d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#103;&#101;&#116;&#32;&#116;&#104;&#101;&#125;&#92;&#116;&#101;&#120;&#116;&#98;&#102;&#123;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#44;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#125;&#46;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"359\" style=\"vertical-align: -15px;\" \/><\/p>\n<p id=\"fs-id1169596378042\">What do you observe about the products?<\/p>\n<p id=\"fs-id1169596378045\">The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.<\/p>\n<p id=\"fs-id1169596303868\">Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.<\/p>\n<p id=\"fs-id1169596389958\">The product of conjugates is always of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fcca822bd1f88970bae37422762ae16f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: 0px;\" \/>. This is called a difference of squares.<\/p>\n<p id=\"fs-id1169596391251\">This leads to the pattern:<\/p>\n<div id=\"fs-id1169596391023\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Product of Conjugates Pattern<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1169596391028\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> are real numbers,<\/p>\n<p><span id=\"fs-id1169596367551\" data-type=\"media\" data-alt=\"This figure is divided into two sides. On the left side is the following formula: the product of a minus b and a plus b equals a squared minus b squared. On the right side is the same formula labeled: a minus b and a plus b are labeled \u201cconjugates\u201d, the a squared and b squared are labeled squares and the minus sign between the squares is labeled \u201cdifference\u201d. Therefore, the product of two conjugates is called a difference of squares.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_020_img_new.jpg\" alt=\"This figure is divided into two sides. On the left side is the following formula: the product of a minus b and a plus b equals a squared minus b squared. On the right side is the same formula labeled: a minus b and a plus b are labeled \u201cconjugates\u201d, the a squared and b squared are labeled squares and the minus sign between the squares is labeled \u201cdifference\u201d. Therefore, the product of two conjugates is called a difference of squares.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<p>The product is called a difference of squares.<\/p>\n<\/div>\n<p id=\"fs-id1169596566014\">To multiply conjugates, square the first term, square the last term, and write the product as a difference of squares.<\/p>\n<\/div>\n<p id=\"fs-id1169596279072\">Let\u2019s test this pattern with a numerical example.<\/p>\n<table id=\"eip-920\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-38a2b48780fbf045bb82801476f8c841_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>It is the product of conjudgates, so the result will be the difference of two squares.<\/td>\n<td>____ \u2013 ____<\/td>\n<\/tr>\n<tr>\n<td>Square the first term.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-68a96dbc61bbd99302a0c5000d55421d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#117;&#110;&#100;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#113;&#113;&#117;&#97;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Square the last term.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0a8473119f3281eb4aaf4f75eeee4c5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#50;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"62\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c7bda995d962a48a0ce0e87f55c6a435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-662f6ae8c8d225801ecf7f82bc73ef62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>What do you get using the order of operations?<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8bc7148077be600956b7e1bde2dae6af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#57;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"125\" style=\"vertical-align: -22px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169596404577\">Notice, the result is the same!<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596404708\" data-type=\"problem\">\n<p id=\"fs-id1169596404710\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4abcf36d235d10dae7c4cc73c77d8632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596403359\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1169596403364\">First, recognize this as a product of conjugates. The binomials have the same first terms, and the same last terms, and one binomial is a sum and the other is a difference.<\/p>\n<table id=\"eip-id1172188052941\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cIt fits the pattern.\u201d The right column contains the product of x minus 8, in parentheses, and x plus 8, in parentheses. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses. In the second row, the left column says \u201cSquare the first term, x.\u201d The right column contains x squared minus blank. Above this is the general form a squared minus b squared. In the third row, the text on the left says \u201cSquare the last term, 8.\u201d The right column contains the expression x squared minus 8 squared. Above this is the general form a squared minus b squared. In the last row, the text in the left column says \u201cThe product is a difference of squares.\u201d The right column contains the expression x squared minus 64. Above this is the general form a squared minus b squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td>It fits the pattern.<\/td>\n<td><span id=\"eip-id1172188052961\" data-type=\"media\" data-alt=\"The product of x minus 8 and x plus 8. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013a_img_new.jpg\" alt=\"The product of x minus 8 and x plus 8. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the first term, <em data-effect=\"italics\">x<\/em>.<\/td>\n<td><span id=\"eip-id1172188052978\" data-type=\"media\" data-alt=\"x squared minus blank. Above this is the general form a squared minus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013b_img_new.jpg\" alt=\"x squared minus blank. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the last term, 8.<\/td>\n<td><span id=\"eip-id1172188052994\" data-type=\"media\" data-alt=\"x squared minus 8 squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013c_img_new.jpg\" alt=\"x squared minus 8 squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>The product is a difference of squares.<\/td>\n<td><span id=\"eip-id1172188053011\" data-type=\"media\" data-alt=\"x squared minus 64.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013d_img_new.jpg\" alt=\"x squared minus 64.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596382320\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596376846\" data-type=\"exercise\">\n<div id=\"fs-id1169596376848\" data-type=\"problem\">\n<p>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bc09d7b277a800c3a07856453f3458cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596308784\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596569338\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-89af9c0f8fbe8f342f185088c80414cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596568300\" data-type=\"problem\">\n<p id=\"fs-id1169596568302\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e5e52a3577ec735f163e13a993b0bf8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596569446\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596569448\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e06af8bcaa3e7e182901508dcc1554e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596568294\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596568297\" data-type=\"exercise\">\n<div id=\"fs-id1169596569446\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596361165\" data-type=\"problem\">\n<p>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-87f4e3e0a9f1e7bf43c54efb411e3874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596392529\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1169596566851\">Are the binomials conjugates?<\/p>\n<table id=\"eip-id1172187850054\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cIt is the product of conjugates.\u201d The right column contains the product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses. In the second row, the left column says \u201cSquare the first term, 2x.\u201d The right column contains 2x squared minus blank. Above this is the general form a squared minus b squared. In the third row, the text on the left says \u201cSquare the last term, 5.\u201d The right column contains the expression 2x squared minus 5 squared. Above this is the general form a squared minus b squared. In the last row, the text in the left column says \u201cSimplify. The product is a difference of squares.\u201d The right column contains the expression 4x squared minus 25. Above this is the general form a squared minus b squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td>It is the product of conjugates.<\/td>\n<td><span id=\"eip-id1172187850074\" data-type=\"media\" data-alt=\"The product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_014a_img_new.jpg\" alt=\"The product of 2x plus 5 and 2x minus 5. Above this is the general form a minus b, in parentheses, times a plus b, in parentheses.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the first term, 2<em data-effect=\"italics\">x<\/em>.<\/td>\n<td><span id=\"eip-id1172187850091\" data-type=\"media\" data-alt=\"2 x squared minus blank. Above this is the general form a squared minus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_014b_img_new.jpg\" alt=\"2 x squared minus blank. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Square the last term, 5.<\/td>\n<td><span id=\"eip-id1172187850108\" data-type=\"media\" data-alt=\"2 x squared minus 5 squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_014c_img_new.jpg\" alt=\"2 x squared minus 5 squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify. The product is a difference of squares.<\/td>\n<td><span id=\"eip-id1172187850124\" data-type=\"media\" data-alt=\"4 x squared minus 25.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_014d_img_new.jpg\" alt=\"4 x squared minus 25.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596566475\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596566479\" data-type=\"exercise\">\n<div id=\"fs-id1169596555445\" data-type=\"problem\">\n<p id=\"fs-id1169596555447\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-67ebb0b3b20a5b16c4411f356f4bebea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596308819\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596308821\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-41800faa6af8a4f35c7b7ec074cbf243_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"74\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596366001\" data-type=\"problem\">\n<p id=\"fs-id1169596381369\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7740c780253bcb542804f2420540c0c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596404475\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596404477\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2d7a3dbd8ebef181b79b07d42f2b1fb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169596276227\">The binomials in the next example may look backwards \u2013 the variable is in the second term. But the two binomials are still conjugates, so we use the same pattern to multiply them.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596276239\" data-type=\"problem\">\n<p id=\"fs-id1169596276241\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-500728a22065c81e8252a82df9fc9195_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#53;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#53;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596574076\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187955273\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cIt is the product of conjugates.\u201d In the right column is the product of 3 plus 5x and 3 minus 5x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 3 squared minus 5x squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 9 minus 25x squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td>It is the product of conjugates.<\/td>\n<td><span id=\"eip-id1172187955293\" data-type=\"media\" data-alt=\"The product of 3 plus 5 x and 3 minus 5 x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_015a_img_new.jpg\" alt=\"The product of 3 plus 5 x and 3 minus 5 x. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172187955310\" data-type=\"media\" data-alt=\"3 squared minus 5 x squared. Above this is the general form a squared minus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_015b_img_new.jpg\" alt=\"3 squared minus 5 x squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172187955326\" data-type=\"media\" data-alt=\"9 minus 25 x squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_015c_img_new.jpg\" alt=\"9 minus 25 x squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596574039\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596574042\" data-type=\"exercise\">\n<div id=\"fs-id1169596574044\" data-type=\"problem\">\n<p id=\"fs-id1169596574046\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ed45279a023451eae7eb49fb087fcdb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#43;&#52;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#52;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596391449\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596391451\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-23c816dd7f2e37e40cb92bef2328d2a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596393084\" data-type=\"problem\">\n<p id=\"fs-id1169596393086\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3b58edb3467d93003e394847b82dce34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#43;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596455633\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596455635\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6afbaceedec0f459cfe969af2789f5f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169596455654\">Now we\u2019ll multiply conjugates that have two variables.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596387831\" data-type=\"problem\">\n<p id=\"fs-id1169596387833\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-671a66312a6ea4c4395477b7a480cefe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#109;&#45;&#57;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#109;&#43;&#57;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"160\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596370494\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172182437772\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cThis fits the pattern.\u201d In the right column is the product of 5m minus 9n and 5m plus 9n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 5m squared minus 9n squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 25m squared minus 81n squared.\" data-label=\"\">\n<tbody>\n<tr>\n<td>This fits the pattern.<\/td>\n<td><span id=\"eip-id1172182437792\" data-type=\"media\" data-alt=\"5 m minus 9 n and 5 m plus 9 n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_016a_img_new.jpg\" alt=\"5 m minus 9 n and 5 m plus 9 n. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172182437809\" data-type=\"media\" data-alt=\"5 m squared minus 9 n squared. Above this is the general form a squared minus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_016b_img_new.jpg\" alt=\"5 m squared minus 9 n squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172186802474\" data-type=\"media\" data-alt=\"25 m squared minus 81 n squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_016c_img_new.jpg\" alt=\"25 m squared minus 81 n squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596570142\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596570145\" data-type=\"exercise\">\n<div id=\"fs-id1169596570147\" data-type=\"problem\">\n<p id=\"fs-id1169596570149\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7cadc8a6619f5eecf4ce6fadee6f985d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#45;&#55;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#112;&#43;&#55;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596364491\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596364494\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8e5b0ca0dff5f9f08fcbc95fb0717430_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596392882\" data-type=\"problem\">\n<p id=\"fs-id1169596392884\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0e8f3eb5122f8c26676df05a280a0fd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596555078\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596384770\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9f1db314b96c71687613e1c29debcb4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596392876\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596392880\" data-type=\"exercise\">\n<div id=\"fs-id1169596555078\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596360904\" data-type=\"problem\">\n<p id=\"fs-id1169596360906\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1e0a6f80e477c407f84e6373f6d4863d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#100;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#100;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596568683\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187818160\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cThis fits the pattern.\u201d In the right column is the product of cd minus 8 and cd plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression cd squared minus 8 squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains c squared d squared minus 64.\" data-label=\"\">\n<tbody>\n<tr>\n<td>This fits the pattern.<\/td>\n<td><span id=\"eip-id1172187818180\" data-type=\"media\" data-alt=\"The product of c d minus 8 and c d plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_017a_img_new.jpg\" alt=\"The product of c d minus 8 and c d plus 8. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172188053182\" data-type=\"media\" data-alt=\"c d squared minus 8 squared. Above this is the general form a squared minus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_017b_img_new.jpg\" alt=\"c d squared minus 8 squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172188053198\" data-type=\"media\" data-alt=\"c squared d squared minus 64.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_017c_img_new.jpg\" alt=\"c squared d squared minus 64.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596303932\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596303936\" data-type=\"exercise\">\n<div id=\"fs-id1169596303938\" data-type=\"problem\">\n<p id=\"fs-id1169596303940\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a5eccc58584b69114eeedfb336d87d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#121;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596566590\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596566592\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-65598c2d0e8955f6153bf5814f24e52d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596574406\" data-type=\"problem\">\n<p id=\"fs-id1169596574408\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-108325d648ddba7136b2c6f45e5ee8d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#98;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#98;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596366993\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596366995\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1ec8855ea04d75bd1f500bc20a0490f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"71\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596574400\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596574404\" data-type=\"exercise\">\n<div id=\"fs-id1169596366993\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596569315\" data-type=\"problem\">\n<p id=\"fs-id1169596569317\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d03c1293a628a95e5b1ac08a37a2ca8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#123;&#118;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#123;&#118;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"197\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-id1172187628858\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the text in the left column says \u201cThis fits the pattern.\u201d In the right column is the product of 6u squared minus 11v to the fifth power and 68 squared plus 11v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 6u squared, in parentheses, squared, minus 11v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 36u to the fourth power minus 121v to the tenth power.\" data-label=\"\">\n<tbody>\n<tr>\n<td>This fits the pattern.<\/td>\n<td><span id=\"eip-id1172186035863\" data-type=\"media\" data-alt=\"The product of 6 u squared minus 11 v to the fifth power and 6 u squared plus 11 v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_018a_img_new.jpg\" alt=\"The product of 6 u squared minus 11 v to the fifth power and 6 u squared plus 11 v to the fifth power. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172186035880\" data-type=\"media\" data-alt=\"6 u squared, in parentheses, squared, minus 11 v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_018b_img_new.jpg\" alt=\"6 u squared, in parentheses, squared, minus 11 v to the fifth power, in parentheses, squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172186035897\" data-type=\"media\" data-alt=\"36 u to the fourth power minus 121 v to the tenth power.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_018c_img_new.jpg\" alt=\"36 u to the fourth power minus 121 v to the tenth power.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596302415\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596302419\" data-type=\"exercise\">\n<div id=\"fs-id1169596302421\" data-type=\"problem\">\n<p id=\"fs-id1169596302423\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0bd5af317430a067d07813b484b3547e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"180\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<details>\n<summary>Show answer<\/summary>\n<div id=\"fs-id1169596303989\" data-type=\"solution\"><\/div>\n<\/details>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6ed09a6417013191eb4dcf47ecc548e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596304022\" data-type=\"problem\">\n<p id=\"fs-id1169596296112\">Find the product: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2c041105d2afa3988478a56ed8d3d261_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"193\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1169596303217\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596303219\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2a99a2d57f53ed3a738895e8287c700c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#45;&#50;&#53;&#123;&#110;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Recognize and Use the Appropriate Special Product Pattern<\/h1>\n<p id=\"fs-id1169596299459\">We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.<\/p>\n<div id=\"fs-id1169596299465\" data-type=\"note\">\n<div data-type=\"title\">Comparing the Special Product Patterns<\/div>\n<table id=\"eip-391\" class=\"grid\" summary=\"\/\">\n<tbody>\n<tr>\n<td><strong>Binomial Squares<\/strong><\/td>\n<td><strong>Product of Conjugates<\/strong><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1cfbdf72854a6dc63ba0f2ae17f46708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-84329933bdaada5a04696a6506bb6d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a05bb1a6b69ab51f7d68b9cad2326fd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>&#8211; Squaring a binomial<\/td>\n<td>&#8211; Multiplying conjugates<\/td>\n<\/tr>\n<tr>\n<td>&#8211; Product is a <strong>trinomial<\/strong><\/td>\n<td>&#8211; Product is a <strong>binomial<\/strong><\/td>\n<\/tr>\n<tr>\n<td>&#8211; Inner and outer terms with FOIL are <strong>the same.<\/strong><\/td>\n<td>&#8211; Inner and outer terms with FOIL are <strong>opposites.<\/strong><\/td>\n<\/tr>\n<tr>\n<td>&#8211; Middle term is <strong>double the product<\/strong> of the terms.<\/td>\n<td>&#8211; There is <strong>no<\/strong> middle term.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<div id=\"fs-id1169596566225\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596566229\" data-type=\"exercise\">\n<div id=\"fs-id1169596566231\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p>Choose the appropriate pattern and use it to find the product:<\/p>\n<p id=\"fs-id1169596499292\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-834f4efc848e7c65b64ca5db80586153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2ad1e42600709f81c5ce8431d5cee912_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"69\" style=\"vertical-align: -5px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-001bd5e0af0743a5f349cf18b4cb65c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#109;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d951078281db37e1ee7c5bf0d394408b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169596369746\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<ol id=\"fs-id1168746418593\" class=\"circled\" type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-834f4efc848e7c65b64ca5db80586153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/> These are conjugates. They have the same first numbers, and the same last numbers, and one binomial is a sum and the other is a difference. It fits the Product of Conjugates pattern.<br \/>\n<table id=\"eip-id1172186037932\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains the product of 2x minus 3 and 2x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 2x squared minus 3 squared. Above this is the general form a squared minus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 4x squared minus 9.\" data-label=\"\">\n<tbody>\n<tr>\n<td>This fits the pattern.<\/td>\n<td><span id=\"eip-id1172186037952\" data-type=\"media\" data-alt=\"The product of 2 x minus 3 and 2 x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_019a_img_new.jpg\" alt=\"The product of 2 x minus 3 and 2 x plus 3. Above this is the general form a plus b, in parentheses, times a minus b, in parentheses.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172185704714\" data-type=\"media\" data-alt=\"2 x squared minus 3 squared. Above this is the general form a squared minus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_019b_img_new.jpg\" alt=\"2 x squared minus 3 squared. Above this is the general form a squared minus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172185704731\" data-type=\"media\" data-alt=\"4 x squared minus 9.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_019c_img_new.jpg\" alt=\"4 x squared minus 9.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c65b9e9c4c1640438744a050eef5928e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/> We are asked to square a binomial. It fits the <strong data-effect=\"bold\">binomial squares<\/strong> pattern.<br \/>\n<table id=\"eip-id1172188053327\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 8x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 8x squared minus 2 times 8x times 5 plus 5 squared. Above this is the general form a squared minus 2 times a times b plus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 64x squared minus 80x plus 25.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172184443423\" data-type=\"media\" data-alt=\"8 x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_021a_img_new.jpg\" alt=\"8 x minus 5, in parentheses, squared. Above this is the general form a minus b, in parentheses, squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172184443439\" data-type=\"media\" data-alt=\"8 x squared minus 2 times 8 x times 5 plus 5 squared. Above this is the general form a squared minus 2 a b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_021b_img_new.jpg\" alt=\"8 x squared minus 2 times 8 x times 5 plus 5 squared. Above this is the general form a squared minus 2 a b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172184443456\" data-type=\"media\" data-alt=\"64 x squared minus 80 x plus 25.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_021c_img_new.jpg\" alt=\"64 x squared minus 80 x plus 25.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-001bd5e0af0743a5f349cf18b4cb65c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#109;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/> Again, we will square a binomial so we use the <strong data-effect=\"bold\">binomial squares<\/strong> pattern.<br \/>\n<table id=\"eip-id1172182439091\" class=\"unnumbered unstyled\" summary=\"This figure has two columns, with written instructions on the left and math on the right. At the top of the figure, the right column contains 6m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared. In the second row, the text in the left column says \u201cUse the pattern.\u201d The right column contains the expression 6m squared plus 2 times 6m times 7 plus 7 squared. Above this is the general form a squared plus 2 times a times b plus b squared. In the bottom row, the text in the left column says \u201cSimplify.\u201d The right column contains 36m squared plus 84m plus 49.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1172187673200\" data-type=\"media\" data-alt=\"6 m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_022a_img_new.jpg\" alt=\"6 m plus 7, in parentheses, squared. Above this is the general form a plus b, in parentheses, squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Use the pattern.<\/td>\n<td><span id=\"eip-id1172187673216\" data-type=\"media\" data-alt=\"6 m squared plus 2 times 6 m times 7 plus 7 squared. Above this is the general form a squared plus 2 a b plus b squared.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_022b_img_new.jpg\" alt=\"6 m squared plus 2 times 6 m times 7 plus 7 squared. Above this is the general form a squared plus 2 a b plus b squared.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><span id=\"eip-id1172187673233\" data-type=\"media\" data-alt=\"36 m squared plus 84 m plus 49.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_022c_img_new.jpg\" alt=\"36 m squared plus 84 m plus 49.\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d951078281db37e1ee7c5bf0d394408b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/> This product does not fit the patterns, so we will use FOIL.<br \/>\n<table id=\"eip-569\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d951078281db37e1ee7c5bf0d394408b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Use FOIL.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f7e9fcf22bb6a2b891eb1b3d9d8b8631_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#120;&#45;&#51;&#54;&#120;&#45;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"174\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1bc5e04dfa97b9d0064077bfd133a1d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#49;&#120;&#45;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"125\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596566225\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1169596566229\" data-type=\"exercise\">\n<div id=\"fs-id1169596566231\" data-type=\"problem\">\n<p id=\"fs-id1169596566233\">Choose the appropriate pattern and use it to find the product:<\/p>\n<p id=\"fs-id1168743981551\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-71424103589302253a54cbed32b13071_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#98;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#98;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-464aee29ee8dd3429d1cfc140a12338f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#112;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4c345bbd1319914d9e574f8a03bbc09e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-04a893173da8dff09cd5a8e9c8f79ded_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#114;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#114;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169596320033\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596320035\">a) FOIL; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-05bff08383e0bf45bb73f85986a173e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#55;&#55;&#98;&#45;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -2px;\" \/> b) Binomial Squares; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-afebec9210e4d15ba9da87863f392dec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#50;&#112;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/> c) Binomial Squares; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7993739088453adf9654bac6e7c8b714_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#121;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/> d) Product of Conjugates; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-445c0c14b1ae0638eaceceb22c8e7b0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1169596567460\" data-type=\"problem\">\n<p id=\"fs-id1169596567463\">Choose the appropriate pattern and use it to find the product:<\/p>\n<p id=\"fs-id1168746404828\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-cc5c7e5121a038e94ca009de9e2e4832_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d8f5ab687361994c8169fc2810b7b16c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5b8de97089b547b1f50ab4525e58cccd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b926cb3647453b418236342819b62009_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1169596573993\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1169596573995\">a) Binomial Squares; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-52cc53827d98c39eb294a7f2e07f7071_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#52;&#120;&#43;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"125\" style=\"vertical-align: -2px;\" \/> b) Product of Conjugates; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b696c75cf014e5f253b533b9fce13f95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/> c) FOIL; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-cfad41ec954bdcdb7bdcd16a7877753c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#57;&#120;&#43;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"124\" style=\"vertical-align: -2px;\" \/> d) Binomial Squares; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0a93d37c279b13794c059a35a3e5a719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#110;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>Access these online resources for additional instruction and practice with special products:<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596291548\" class=\"media-2\" data-type=\"note\">\n<ul id=\"fs-id1169596291556\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/25Specialprod\">Special Products<\/a><\/li>\n<\/ul>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1169596457039\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Binomial Squares Pattern<\/strong>\n<ul id=\"fs-id1169596457051\" data-bullet-style=\"open-circle\">\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfaed44949cf9cfbeb3445de33aabd3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"25\" style=\"vertical-align: -4px;\" \/> are real numbers,<br \/>\n<span id=\"fs-id1168744116889\" data-type=\"media\" data-alt=\"No Alt Text\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_013_img_new.jpg\" alt=\"No Alt Text\" data-media-type=\"image\/jpeg\" \/><\/span><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1cfbdf72854a6dc63ba0f2ae17f46708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a05bb1a6b69ab51f7d68b9cad2326fd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>To square a binomial: square the first term, square the last term, double their product.<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Product of Conjugates Pattern<\/strong>\n<ul data-bullet-style=\"open-circle\">\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bfaed44949cf9cfbeb3445de33aabd3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"25\" style=\"vertical-align: -4px;\" \/> are real numbers,<br \/>\n<span id=\"fs-id1168746511136\" data-type=\"media\" data-alt=\"No Alt Text\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/accessibilitytoolkit\/wp-content\/uploads\/sites\/333\/2021\/03\/CNX_ElemAlg_Figure_06_04_023_img_new.jpg\" alt=\"No Alt Text\" data-media-type=\"image\/jpeg\" \/><\/span><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-84329933bdaada5a04696a6506bb6d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>The product is called a difference of squares.<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">To multiply conjugates:<\/strong>\n<ul id=\"fs-id1169596570314\" data-bullet-style=\"open-circle\">\n<li><strong data-effect=\"bold\">square the first term square the last term<\/strong> write it as a difference of squares<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1169596373142\">\n<dt>conjugate pair<\/dt>\n<dd id=\"fs-id1169596568794\">A conjugate pair is two binomials of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-84a71b6db866fdb7531d99a893cd59f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/>; the pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.<\/dd>\n<\/dl>\n<\/div>\n<h1>Practice Makes Perfect<\/h1>\n<h2 id=\"fs-id1169596570340\">Square a Binomial Using the Binomial Squares Pattern<\/h2>\n<p id=\"fs-id1168746325019\">In the following exercises, square each binomial using the Binomial Squares Pattern.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">1. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-880c4af710d86f2f7ad9b31b4e56c7fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">2. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-08c387ecac46d590b2780872b2b691c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#119;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">3. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d4719c722fff60e92590831fce7aa8a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"75\" style=\"vertical-align: -17px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">4. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3b136a38288bfb7adda7e0f0d57ebf55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"74\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">5. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c9ab111836a720bda969b4aa41ffe0c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">6. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-156303236dd24f509b0289caae4b47b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"58\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">7. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-654bbfec32163710abb4a1d570d7edab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#49;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">8. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a2f991b138e56244214379274ba55676_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#49;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">9. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-12c9e3330bc8830c08d965920fe04eff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#97;&#43;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">10. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-bdfa77ff39a2965fe6cf996f9d2f5264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#100;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">11. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ab2374ea8fd840c23e1f32412d5fb2a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#122;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"82\" style=\"vertical-align: -17px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">12. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d1d6947c431d03017e81b042efe7b1d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#113;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"82\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">13. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b58402b47b15379a3bfd678f77fd1417_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#51;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">14. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-611965b85a4c9619d830002fc6f0ff44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">15. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fbe77b26fca87cdfc0fe530ec3649b8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#120;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"97\" style=\"vertical-align: -17px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">16. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-46972c1ee5b3145e0e678c1933836404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#55;&#125;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"97\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">17. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9eb2376df7570f496de3f3fbd929ea35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">18. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5248ff65b20af3867e896302cad8c7ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">19. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-16d7dd393da3a4820532fe51ba3f69fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"77\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">20. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a273a7442457c5093c8a0f4324640547_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"text-align: initial; background-color: initial;\">In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.<\/span><\/p>\n<h2>Multiply Conjugates Using the Product of Conjugates Pattern<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">21. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6600ad564c1884cb4422e77cc6a66b0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">22. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-8a99338a9fbe0225a1ac435b397be3ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">23. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7533ff9fb8dae765783f2da7849d9669_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"133\" style=\"vertical-align: -17px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">24. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6dccd0d67d1d1199086f9feb9030edb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"138\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">25. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5341b76eaa9962e35039c5c5d09e728c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#106;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#106;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">26. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-5ecc824e2fc55baf13e4e92d0ae2e0ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#107;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#107;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">27. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-02528cb5cc2d9083233d00e1620009e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">28. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1995ae9214917453a83726cdb80b069f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#107;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#107;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">29. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-af3e3e440de1017ad7eb49acc911a540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#45;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#43;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">30. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fc10880b79bc7b849abfbf9eae6848ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">31. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-9f1fcbf476e0e811f3807064591b83df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#45;&#54;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#54;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">32. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2fb1a7b3e3891cf04517b8011efaf701_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#51;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#43;&#51;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">33. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-eb4d4dfecf381812f67bbf08871e15bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#119;&#43;&#49;&#48;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#119;&#45;&#49;&#48;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">34. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-286195b24cdae56772b032fd099def5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#45;&#50;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#99;&#43;&#50;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">35. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-827a0169c5ee39b6a3a75aac01c3cce9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -17px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">36. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-016e463203cb403485430826f17a3529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"171\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">37. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f44fc40d410c580abaafbb7dfba17010_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#121;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#121;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">38. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3e8c1717020ba8d3088954481f1fccf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#98;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#98;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 50%; height: 14px;\">39. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-64ae0976d3a7d92e609acc4b1f07463e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#115;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#115;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"152\" style=\"vertical-align: -17px;\" \/><\/td>\n<td style=\"width: 50%; height: 14px;\">40. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-abd6dce8e6fd47cbfc41c3a27b40efcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#118;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#117;&#118;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"157\" style=\"vertical-align: -17px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">41. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-f4a7321d018b1bd75b60524b1c14efca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#123;&#110;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#43;&#52;&#123;&#110;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"193\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">42. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ae037cde7f0aa7d7215f23bbaaee2d16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"180\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">43. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-48da09faf6b4136e682cfadefeb5e692_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#123;&#110;&#125;&#94;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"211\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">44. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-73bca837fc18bac3c2a9b95511a702b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#49;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#49;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"211\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1169596376643\"><strong data-effect=\"bold\">In the following exercises, find each product.<\/strong><\/p>\n<h2>Recognize and Use the Appropriate Special Product Pattern<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"fs-id1169596376650\" data-type=\"exercise\">\n<div id=\"fs-id1169596376652\" data-type=\"problem\">\n<p>45.<\/p>\n<p>a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-010754e05eeaf2e133304ec096739027_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#114;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596568003\" data-type=\"exercise\">\n<div id=\"fs-id1169596568005\" data-type=\"problem\">\n<p>b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1af6419091d72a823bb4c200caaa9879_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-26810c6033f2bbd628b69cd33d4d3e18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#55;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1becfc3c29342c37cd6f9ed7963b469c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">46.<\/p>\n<p>a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7cedef0c8aaf6a2be62c8da925ebe7a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#112;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b16ad6f040394f8967bc6b5d9bc8888b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#116;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b18968046794dcfefc0d58d65558b601_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6c804d0967eedc3199dcbad4b8bd5184_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">47.<\/p>\n<p>a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2ee1303216ae8627c21847dde21969cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#53;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<p>b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0b4598a3aef0eb2119678554f5131afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#45;&#56;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"86\" style=\"vertical-align: -7px;\" \/><\/p>\n<p>c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7be131f431b23bc0aa25e63bb1cb78e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#112;&#43;&#56;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-ef31476c6bb2df7b20987d0469d503eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"139\" style=\"vertical-align: -7px;\" \/><\/td>\n<td style=\"width: 50%;\">48.<\/p>\n<p>a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-39dcc31c0bc5a1c89073d03b5f342916_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#53;&#125;&#45;&#55;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"76\" style=\"vertical-align: -7px;\" \/><\/p>\n<p>b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7530eb3031f97d7ba7bfba6221fa16a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#120;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"147\" style=\"vertical-align: -7px;\" \/><\/p>\n<p>c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-62551eb8545ee4c0d0bede50f4a785e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#54;&#125;&#43;&#123;&#115;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#114;&#125;&#94;&#123;&#54;&#125;&#45;&#123;&#115;&#125;&#94;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"139\" style=\"vertical-align: -7px;\" \/><\/p>\n<p>d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0e7c317815c665702604dc37007c5304_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#43;&#50;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"78\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 data-type=\"title\">Everyday Math<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1169596364200\">49. <strong data-effect=\"bold\">Mental math<\/strong> You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4c32d8ad9a2e6bc8953de3bd8e375cb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"47\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<ol id=\"fs-id1168746270495\" class=\"circled\" type=\"a\">\n<li>Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-da4602e17156a0be1ef1f28f00540eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#48;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/> by using the binomial squares pattern, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1cfbdf72854a6dc63ba0f2ae17f46708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"184\" style=\"vertical-align: -4px;\" \/>.<\/li>\n<li>Square 65 without using a calculator.<\/li>\n<li>Which way is easier for you? Why?<\/li>\n<\/ol>\n<\/td>\n<td style=\"width: 50%;\">\n<p id=\"fs-id1169596568354\">50.<strong data-effect=\"bold\"> Mental math<\/strong> You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-77671d697697a34d22b044c54088a12d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: 0px;\" \/> and 53 as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0c2cce6d13a85dde1dd8a3d05339b503_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"48\" style=\"vertical-align: -2px;\" \/>.<\/p>\n<ol id=\"fs-id1168744264659\" class=\"circled\" type=\"a\">\n<li>Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-d5d5f8e124b28ac7c5c441b77d1e83cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#48;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#48;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/> by using the product of conjugates pattern, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-84329933bdaada5a04696a6506bb6d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -4px;\" \/>.<\/li>\n<li>Multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-867a28d539a4b59d2e0858c0dad5109d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#55;&#92;&#99;&#100;&#111;&#116;&#32;&#53;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/> without using a calculator.<\/li>\n<li>Which way is easier for you? Why?<\/li>\n<\/ol>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 data-type=\"title\">Writing Exercises<\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">52. Why does <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fcf4905322cd42d5aefebd64aa6a523b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"58\" style=\"vertical-align: -4px;\" \/> result in a trinomial, but <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b62cb51f5db877a2930ac0090238692d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/> result in a binomial?<\/td>\n<td style=\"width: 50%;\">51. How do you decide which pattern to use?<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">54. Use the order of operations to show that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1574144c506f34e85d0ea9a4e3484832_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"59\" style=\"vertical-align: -4px;\" \/> is 64, and then use that numerical example to explain why <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-4703fc4c7fe9b7aaf41216114a7dbd9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#110;&#101;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"136\" style=\"vertical-align: -4px;\" \/>.<\/td>\n<td style=\"width: 50%;\">\n<div id=\"fs-id1169596387451\" data-type=\"exercise\">\n<div id=\"fs-id1169596387453\" data-type=\"problem\">\n<p id=\"fs-id1169596555462\">53. Marta did the following work on her homework paper:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-101755f15638f1df4c512b5ff48e5cbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#32;&#92;&#92;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#32;&#92;&#92;&#57;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"60\" style=\"vertical-align: -27px;\" \/><\/p>\n<p id=\"fs-id1169596555524\">Explain what is wrong with Marta\u2019s work.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1169596555535\" data-type=\"exercise\">\n<div id=\"fs-id1169596555537\" data-type=\"problem\">\n<p id=\"fs-id1169596555540\">\n<\/div>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Answers<\/h1>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">1. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b51add5c0247816bd7d8b1359801b111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#113;&#43;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">3. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-1811fa22b8a4d7dd83e0c959495045b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"95\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">5. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-b9c80d7e6c576337f37eb95cdb76ee02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#121;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">7. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-6ab32cd2a6198821a617d180d2ffacf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#54;&#112;&#43;&#49;&#54;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">9. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7e38999a27644cc3fa5182c30dd74f1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#48;&#97;&#43;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">11. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-925a3494b68b449a4f9036f8aa6f27ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#122;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"111\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 34px;\">\n<td style=\"width: 50%; height: 34px;\">13. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a45104700e1a3f1b961d32b74279349b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#121;&#122;&#43;&#57;&#123;&#122;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 34px;\">15. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c53aa5cffa9c8c86af48c6dcedf614c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#52;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#54;&#125;&#120;&#121;&#43;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#49;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"160\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">17. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e6e6f8cc2da21a4dfbe359e1cafb3cdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#117;&#125;&#94;&#123;&#52;&#125;&#43;&#57;&#48;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"131\" style=\"vertical-align: -2px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">19. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-c144bb025db29c3959ce13f01d519ce4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#112;&#125;&#94;&#123;&#54;&#125;&#45;&#52;&#56;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">21. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2e4da13e93f9501f3280fe488ca2cea9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: 0px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">23. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-62705b918f7a8196071f8499391278ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#51;&#54;&#125;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"57\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">25. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-31bd689ede5e47b3e009f537f15c2340_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#106;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">27. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-99ca19c4f8177e6f77d1076bba023655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">29. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-44283491eae8b210b598085ee9d58dc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#57;&#45;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">31. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-7a4faec00c8876d4b0f5da4a51848fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#45;&#51;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">33. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2fe29ffdd9b27920592af9cca58d1147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#119;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">35. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-0c1e91fc30448d9350a5e2136ca1428f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#50;&#53;&#125;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"76\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">37. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-495cdeb1c778fc1fc10bd1d1506a6f43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">39. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-892122f72ff42f1be2359cbb0287efd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#45;&#92;&#100;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"74\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">41. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-3edd9b1499fb7bb3cd32be5de79be717_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#109;&#125;&#94;&#123;&#54;&#125;&#45;&#49;&#54;&#123;&#110;&#125;&#94;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: -1px;\" \/><\/td>\n<td style=\"width: 50%; height: 16px;\">43. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-2521f4f31d63b41b865ac24329ed10be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#50;&#53;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#45;&#54;&#52;&#123;&#110;&#125;&#94;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 70px;\">\n<td style=\"width: 50%; height: 70px;\">45. a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-69be7f3261c35d68470e7fc5f67b7414_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#56;&#114;&#43;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"122\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-e441c3c9b25263613ebc65dd9957d36b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -4px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fa6191c772c53349b70e946b5ce304ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#56;&#97;&#98;&#45;&#55;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"128\" style=\"vertical-align: -1px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a0b519e3726346b19593c30d9ab80b04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#107;&#43;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -2px;\" \/><\/td>\n<td style=\"width: 50%; height: 70px;\">47. a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-feb9cd0e69be586bb66a90478f187788_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#49;&#48;&#125;&#45;&#123;&#121;&#125;&#94;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-fe5f1f9de54421e4d77156ed9c95eed9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#54;&#125;&#45;&#49;&#54;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#110;&#43;&#54;&#52;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"154\" style=\"vertical-align: -2px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-a34ecd40d87663fb01ce6d84b39c9137_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#52;&#112;&#113;&#43;&#54;&#52;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"155\" style=\"vertical-align: -4px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/introalgebra\/wp-content\/ql-cache\/quicklatex.com-865eaa5f7b543726e9aacb39f0f0294d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#114;&#125;&#94;&#123;&#53;&#125;&#43;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#123;&#115;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#115;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"161\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">49. a) 4,225 b) 4,225 c) Answers will vary.<\/td>\n<td style=\"width: 50%; height: 16px;\">51. Answers will vary.<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"width: 50%; height: 16px;\">53. Answers will vary.<\/td>\n<td style=\"width: 50%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h1>Attributions<\/h1>\n<p>This chapter has been adapted from \u201cSpecial Products\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em> (OpenStax)<\/a> by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.<\/p>\n<p><!-- pb_fixme --><\/p>\n","protected":false},"author":90,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2200","chapter","type-chapter","status-publish","hentry"],"part":2016,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2200","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":2,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2200\/revisions"}],"predecessor-version":[{"id":2521,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2200\/revisions\/2521"}],"part":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/parts\/2016"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapters\/2200\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/media?parent=2200"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2200"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/contributor?post=2200"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/introalgebra\/wp-json\/wp\/v2\/license?post=2200"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}