{"id":1379,"date":"2021-12-02T19:37:44","date_gmt":"2021-12-03T00:37:44","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/7-5-factoring-special-products\/"},"modified":"2023-08-30T18:32:25","modified_gmt":"2023-08-30T22:32:25","slug":"factoring-special-products","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/chapter\/factoring-special-products\/","title":{"raw":"7.5 Factoring Special Products","rendered":"7.5 Factoring Special Products"},"content":{"raw":"Now transition from multiplying special products to factoring special products. If you can recognize them, you can save a lot of time. The following is a list of these special products (note that a2 <\/sup>+ b2<\/sup> cannot be factored):\r\n

[latex]\\begin{array}{lll}\r\na^2-b^2&=&(a+b)(a-b) \\\\\r\n(a+b)^2&=&a^2+2ab+b^2 \\\\\r\n(a-b)^2&=&a^2-2ab+b^2 \\\\\r\na^3-b^3&=&(a-b)(a^2+ab+b^2) \\\\\r\na^3+b^3&=&(a+b)(a^2-ab+b^2) \\\\\r\n\\end{array}[\/latex]<\/p>\r\nThe challenge is therefore in recognizing the special product.\r\n

\r\n

Example 7.5.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor [latex]x^2 - 36[\/latex].\r\n\r\nThis is a difference of squares. [latex](x - 6)(x + 6)[\/latex] is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor [latex]x^2 - 6x + 9[\/latex].\r\n\r\nThis is a perfect square. [latex](x - 3)(x - 3)[\/latex] or [latex](x - 3)^2[\/latex] is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor [latex]x^2 + 6x + 9[\/latex].\r\n\r\nThis is a perfect square. [latex](x + 3)(x + 3)[\/latex] or [latex](x + 3)^2[\/latex] is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.4<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor [latex]4x^2 + 20xy + 25y^2[\/latex].\r\n\r\nThis is a perfect square. [latex](2x + 5y)(2x + 5y)[\/latex] or [latex](2x + 5y)^2[\/latex] is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.5<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor [latex]m^3 - 27[\/latex].\r\n\r\nThis is a difference of cubes. [latex](m - 3)(m^2 + 3m + 9)[\/latex] is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.6<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor [latex]125p^3 + 8r^3[\/latex].\r\n\r\nThis is a difference of cubes. [latex](5p + 2r)(25p^2 - 10pr + 4r^2)[\/latex] is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n

Questions<\/h1>\r\nFactor each of the following polynomials.\r\n
    \r\n \t
  1. [latex]r^2-16[\/latex]<\/li>\r\n \t
  2. [latex]x^2-9[\/latex]<\/li>\r\n \t
  3. [latex]v^2-25[\/latex]<\/li>\r\n \t
  4. [latex]x^2-1[\/latex]<\/li>\r\n \t
  5. [latex]p^2-4[\/latex]<\/li>\r\n \t
  6. [latex]4v^2-1[\/latex]<\/li>\r\n \t
  7. [latex]3x^2-27[\/latex]<\/li>\r\n \t
  8. [latex]5n^2-20[\/latex]<\/li>\r\n \t
  9. [latex]16x^2-36[\/latex]<\/li>\r\n \t
  10. [latex]125x^2+45y^2[\/latex]<\/li>\r\n \t
  11. [latex]a^2-2a+1[\/latex]<\/li>\r\n \t
  12. [latex]k^2+4k+4[\/latex]<\/li>\r\n \t
  13. [latex]x^2+6x+9[\/latex]<\/li>\r\n \t
  14. [latex]n^2-8n+16[\/latex]<\/li>\r\n \t
  15. [latex]25p^2-10p+1[\/latex]<\/li>\r\n \t
  16. [latex]x^2+2x+1[\/latex]<\/li>\r\n \t
  17. [latex]25a^2+30ab+9b^2[\/latex]<\/li>\r\n \t
  18. [latex]x^2+8xy+16y^2[\/latex]<\/li>\r\n \t
  19. [latex]8x^2-24xy+18y^2[\/latex]<\/li>\r\n \t
  20. [latex]20x^2+20xy+5y^2[\/latex]<\/li>\r\n \t
  21. [latex]8-m^3[\/latex]<\/li>\r\n \t
  22. [latex]x^3+64[\/latex]<\/li>\r\n \t
  23. [latex]x^3-64[\/latex]<\/li>\r\n \t
  24. [latex]x^3+8[\/latex]<\/li>\r\n \t
  25. [latex]216-u^3[\/latex]<\/li>\r\n \t
  26. [latex]125x^3-216[\/latex]<\/li>\r\n \t
  27. [latex]125a^3-64[\/latex]<\/li>\r\n \t
  28. [latex]64x^3-27[\/latex]<\/li>\r\n \t
  29. [latex]64x^3+27y^3[\/latex]<\/li>\r\n \t
  30. [latex]32m^3-108n^3[\/latex]<\/li>\r\n<\/ol>\r\nAnswer Key 7.5<\/a>","rendered":"

    Now transition from multiplying special products to factoring special products. If you can recognize them, you can save a lot of time. The following is a list of these special products (note that a2 <\/sup>+ b2<\/sup> cannot be factored):<\/p>\n

    [latex]\\begin{array}{lll} a^2-b^2&=&(a+b)(a-b) \\\\ (a+b)^2&=&a^2+2ab+b^2 \\\\ (a-b)^2&=&a^2-2ab+b^2 \\\\ a^3-b^3&=&(a-b)(a^2+ab+b^2) \\\\ a^3+b^3&=&(a+b)(a^2-ab+b^2) \\\\ \\end{array}[\/latex]<\/p>\n

    The challenge is therefore in recognizing the special product.<\/p>\n

    \n
    \n

    Example 7.5.1<\/p>\n<\/header>\n

    \n

    Factor [latex]x^2 - 36[\/latex].<\/p>\n

    This is a difference of squares. [latex](x - 6)(x + 6)[\/latex] is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.2<\/p>\n<\/header>\n

    \n

    Factor [latex]x^2 - 6x + 9[\/latex].<\/p>\n

    This is a perfect square. [latex](x - 3)(x - 3)[\/latex] or [latex](x - 3)^2[\/latex] is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.3<\/p>\n<\/header>\n

    \n

    Factor [latex]x^2 + 6x + 9[\/latex].<\/p>\n

    This is a perfect square. [latex](x + 3)(x + 3)[\/latex] or [latex](x + 3)^2[\/latex] is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.4<\/p>\n<\/header>\n

    \n

    Factor [latex]4x^2 + 20xy + 25y^2[\/latex].<\/p>\n

    This is a perfect square. [latex](2x + 5y)(2x + 5y)[\/latex] or [latex](2x + 5y)^2[\/latex] is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.5<\/p>\n<\/header>\n

    \n

    Factor [latex]m^3 - 27[\/latex].<\/p>\n

    This is a difference of cubes. [latex](m - 3)(m^2 + 3m + 9)[\/latex] is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.6<\/p>\n<\/header>\n

    \n

    Factor [latex]125p^3 + 8r^3[\/latex].<\/p>\n

    This is a difference of cubes. [latex](5p + 2r)(25p^2 - 10pr + 4r^2)[\/latex] is the solution.<\/p>\n<\/div>\n<\/div>\n

    Questions<\/h1>\n

    Factor each of the following polynomials.<\/p>\n

      \n
    1. [latex]r^2-16[\/latex]<\/li>\n
    2. [latex]x^2-9[\/latex]<\/li>\n
    3. [latex]v^2-25[\/latex]<\/li>\n
    4. [latex]x^2-1[\/latex]<\/li>\n
    5. [latex]p^2-4[\/latex]<\/li>\n
    6. [latex]4v^2-1[\/latex]<\/li>\n
    7. [latex]3x^2-27[\/latex]<\/li>\n
    8. [latex]5n^2-20[\/latex]<\/li>\n
    9. [latex]16x^2-36[\/latex]<\/li>\n
    10. [latex]125x^2+45y^2[\/latex]<\/li>\n
    11. [latex]a^2-2a+1[\/latex]<\/li>\n
    12. [latex]k^2+4k+4[\/latex]<\/li>\n
    13. [latex]x^2+6x+9[\/latex]<\/li>\n
    14. [latex]n^2-8n+16[\/latex]<\/li>\n
    15. [latex]25p^2-10p+1[\/latex]<\/li>\n
    16. [latex]x^2+2x+1[\/latex]<\/li>\n
    17. [latex]25a^2+30ab+9b^2[\/latex]<\/li>\n
    18. [latex]x^2+8xy+16y^2[\/latex]<\/li>\n
    19. [latex]8x^2-24xy+18y^2[\/latex]<\/li>\n
    20. [latex]20x^2+20xy+5y^2[\/latex]<\/li>\n
    21. [latex]8-m^3[\/latex]<\/li>\n
    22. [latex]x^3+64[\/latex]<\/li>\n
    23. [latex]x^3-64[\/latex]<\/li>\n
    24. [latex]x^3+8[\/latex]<\/li>\n
    25. [latex]216-u^3[\/latex]<\/li>\n
    26. [latex]125x^3-216[\/latex]<\/li>\n
    27. [latex]125a^3-64[\/latex]<\/li>\n
    28. [latex]64x^3-27[\/latex]<\/li>\n
    29. [latex]64x^3+27y^3[\/latex]<\/li>\n
    30. [latex]32m^3-108n^3[\/latex]<\/li>\n<\/ol>\n

      Answer Key 7.5<\/a><\/p>\n","protected":false},"author":90,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1379","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":1369,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1379","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":2,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1379\/revisions"}],"predecessor-version":[{"id":2126,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1379\/revisions\/2126"}],"part":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/parts\/1369"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapters\/1379\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1379"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/chapter-type?post=1379"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1379"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}