{"id":2269,"date":"2020-08-21T19:42:42","date_gmt":"2020-08-21T23:42:42","guid":{"rendered":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/review\/"},"modified":"2025-09-22T19:26:26","modified_gmt":"2025-09-22T23:26:26","slug":"review","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/review\/","title":{"raw":"8.7 Chapter Review","rendered":"8.7 Chapter Review"},"content":{"raw":"

[latexpage]<\/h3>\n

Review Exercises<\/h1>\n

Identify Polynomials, Monomials, Binomials and Trinomials<\/h2>\n

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.<\/p>\n\n\n\n\n
1.\n\na) \\({a}^{2}-{b}^{2}\\)\nb) \\(24{d}^{3}\\)\nc) \\({x}^{2}+8x-10\\)\nd) \\({m}^{2}{n}^{2}-2mn+6\\)\ne) \\(7{y}^{3}+{y}^{2}-2y-4\\)<\/td>\n2.\n\na) \\(11{c}^{4}-23{c}^{2}+1\\)\nb) \\(9{p}^{3}+6{p}^{2}-p-5\\)\nc) \\(\\frac{3}{7}x+\\frac{5}{14}\\)\nd) 10\ne) \\(2y-12\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Determine the Degree of Polynomials<\/h2>\n

In the following exercises, determine the degree of each polynomial.<\/strong><\/p>\n\n\n\n\n
3.\n\na) \\(5{p}^{3}-8{p}^{2}+10p-4\\)\n\nb) \\(-20{q}^{4}\\)\n\nc) \\({x}^{2}+6x+12\\)\n\nd) \\(23{r}^{2}{s}^{2}-4rs+5\\)\n\ne)\u00a0 100<\/td>\n4.\n\na) \\(3{x}^{2}+9x+10\\)\n\nb) \\(14{a}^{2}bc\\)\n\nc) \\(6y+1\\)\n\nd) \\({n}^{3}-4{n}^{2}+2n-8\\)\n\ne) \\(-19\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Add and Subtract Monomials<\/h2>\n

In the following exercises, add or subtract the monomials.<\/p>\n\n\n\n\n\n\n\n
5. \\(-14k+19k\\)<\/td>\n6. \\({\\phantom{\\rule{0.2em}{0ex}}\\text{5y}}^{\\text{3}}+8{y}^{3}\\)<\/td>\n<\/tr>\n
7. \\(-9c-18c\\)<\/td>\n8. \\(12q-\\left(-6q\\right)\\)<\/td>\n<\/tr>\n
9. \\(3{m}^{2}+7{n}^{2}-3{m}^{2}\\)<\/td>\n10. \\(\\text{12x}-4y-9x\\)<\/td>\n<\/tr>\n
11. \\(\\text{13a}+b\\)<\/td>\n12. \\(6{x}^{2}y-4x+8x{y}^{2}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Add and Subtract Polynomials<\/h2>\n

In the following exercises, add or subtract the polynomials.<\/p>\n\n\n\n\n\n\n
13. \\(\\left(9{p}^{2}-5p+3\\right)+\\left(4{p}^{2}-4\\right)\\)<\/td>\n14. \\(\\left(5{x}^{2}+12x+1\\right)+\\left(6{x}^{2}-8x+3\\right)\\)<\/td>\n<\/tr>\n
15. \\(\\left(7{y}^{2}-8y\\right)-\\left(y-4\\right)\\)<\/td>\n16. \\(\\left(10{m}^{2}-8m-1\\right)-\\left(5{m}^{2}+m-2\\right)\\)<\/td>\n<\/tr>\n
17. Find the sum of \\(\\left({a}^{2}+6a+9\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\left(5{a}^{3}-7\\right)\\)<\/td>\n18. Subtract\n\\(\\left(3{s}^{2}+10\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{from}\\phantom{\\rule{0.2em}{0ex}}\\left(15{s}^{2}-2s+8\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Evaluate a Polynomial for a Given Value of the Variable<\/h2>\n

In the following exercises, evaluate each polynomial for the given value.<\/p>\n\n\n\n\n\n
\n

19. Evaluate \\(10-12x\\) when:<\/p>\na) \\(x=3\\)\n\nb) \\(x=0\\)\n\nc) \\(x=-1\\)<\/td>\n

\n

20. Evaluate \\(3{y}^{2}-y+1\\) when:<\/p>\na) \\(y=5\\)\n\nb) \\(y=-1\\)\n\nc) \\(y=0\\)<\/td>\n<\/tr>\n

21. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p<\/em> dollars each is given by the polynomial \\(-4{p}^{2}+460p\\). Find the revenue received when \\(p=75\\) dollars.<\/td>\n22. Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial \\(-16{t}^{2}+200\\) gives the height of a stone \\(t\\) seconds after it is dropped from the cliff. Find the height after \\(t=3\\) seconds.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Multiply Monomials<\/h2>\n

In the following exercises, multiply the monomials.<\/p>\n\n\n\n\n\n
23. \\(\\left(-9{n}^{7}\\right)\\left(-16n\\right)\\)<\/td>\n24. \\(\\left(-15{x}^{2}\\right)\\left(6{x}^{4}\\right)\\)<\/td>\n<\/tr>\n
25. \\(\\left(\\frac{5}{9}a{b}^{2}\\right)\\left(27a{b}^{3}\\right)\\)<\/td>\n26. \\(\\left(7{p}^{5}{q}^{3}\\right)\\left(8p{q}^{9}\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Multiply a Polynomial by a Monomial<\/h2>\n

In the following exercises, multiply.<\/p>\n\n\n\n\n\n\n\n\n
27. \\(-4\\left(y+13\\right)\\)<\/td>\n28. \\(7\\left(a+9\\right)\\)<\/td>\n<\/tr>\n
29. \\(p\\left(p+3\\right)\\)<\/td>\n30. \\(-5\\left(r-2\\right)\\)<\/td>\n<\/tr>\n
31. \\(-6u\\left(2u+7\\right)\\)<\/td>\n32. \\(-m\\left(m+15\\right)\\)<\/td>\n<\/tr>\n
33. \\(3{q}^{2}\\left({q}^{2}-7q+6\\right)\\) 3<\/td>\n34. \\(9\\left({b}^{2}+6b+8\\right)\\)<\/td>\n<\/tr>\n
35. \\(\\left(b-4\\right)\\cdot 11\\)<\/td>\n36. \\(\\left(5z-1\\right)z\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Multiply a Binomial by a Binomial<\/h2>\n

In the following exercises, multiply the binomials using: a) the Distributive Property, b) the FOIL method, c) the Vertical Method.<\/p>\n\n\n\n\n
37. \\(\\left(6y-7\\right)\\left(2y-5\\right)\\)<\/td>\n38. \\(\\left(x-4\\right)\\left(x+10\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

In the following exercises, multiply the binomials. Use any method.<\/p>\n\n\n\n\n\n\n\n
39. \\(\\left(y-4\\right)\\left(y-8\\right)\\)<\/td>\n40. \\(\\left(x+3\\right)\\left(x+9\\right)\\)<\/td>\n<\/tr>\n
41. \\(\\left(q+16\\right)\\left(q-3\\right)\\)<\/td>\n42. \\(\\left(p-7\\right)\\left(p+4\\right)\\)<\/td>\n<\/tr>\n
43. \\(\\left({u}^{2}+6\\right)\\left({u}^{2}-5\\right)\\)<\/td>\n44. \\(\\left(5m-8\\right)\\left(12m+1\\right)\\)<\/td>\n<\/tr>\n
45. \\(\\left(8mn+3\\right)\\left(2mn-1\\right)\\)<\/td>\n46. \\(\\left(9x-y\\right)\\left(6x-5\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
\n
<\/div>\n<\/div>\n

Multiply a Trinomial by a Binomial<\/h2>\n

In the following exercises, multiply using a) the Distributive Property, b) the Vertical Method.<\/p>\n\n\n\n\n
47. \\(\\left(3x-4\\right)\\left(6{x}^{2}+x-10\\right)\\)<\/td>\n48. \\(\\left(n+1\\right)\\left({n}^{2}+5n-2\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

In the following exercises, multiply. Use either method.<\/p>\n\n\n\n\n
49. \\(\\left(7m+1\\right)\\left({m}^{2}-10m-3\\right)\\)<\/td>\n50. \\(\\left(y-2\\right)\\left({y}^{2}-8y+9\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Square a Binomial Using the Binomial Squares Pattern<\/h2>\n

In the following exercises, square each binomial using the Binomial Squares Pattern.<\/p>\n\n\n\n\n\n\n
51. \\({\\left(q-15\\right)}^{2}\\)<\/td>\n52. \\({\\left(c+11\\right)}^{2}\\)<\/td>\n<\/tr>\n
53. \\({\\left(8u+1\\right)}^{2}\\)<\/td>\n54. \\({\\left(x+\\frac{1}{3}\\right)}^{2}\\)<\/td>\n<\/tr>\n
55. \\({\\left(4a-3b\\right)}^{2}\\)<\/td>\n56. \\({\\left(3{n}^{3}-2\\right)}^{2}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Multiply Conjugates Using the Product of Conjugates Pattern<\/h2>\n

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.<\/p>\n\n\n\n\n\n\n
57. \\(\\left(y+\\frac{2}{5}\\right)\\left(y-\\frac{2}{5}\\right)\\)<\/td>\n58. \\(\\left(s-7\\right)\\left(s+7\\right)\\)<\/td>\n<\/tr>\n
59. \\(\\left(6-r\\right)\\left(6+r\\right)\\)<\/td>\n60. \\(\\left(12c+13\\right)\\left(12c-13\\right)\\)<\/td>\n<\/tr>\n
61. \\(\\left(5{p}^{4}-4{q}^{3}\\right)\\left(5{p}^{4}+4{q}^{3}\\right)\\)<\/td>\n62. \\(\\left(u+\\frac{3}{4}v\\right)\\left(u-\\frac{3}{4}v\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Recognize and Use the Appropriate Special Product Pattern<\/h2>\n

In the following exercises, find each product.<\/p>\n\n\n\n\n\n\n
63. \\(\\left(6a+11\\right)\\left(6a-11\\right)\\)<\/td>\n64. \\({\\left(3m+10\\right)}^{2}\\)<\/td>\n<\/tr>\n
65. \\({\\left({c}^{4}+9d\\right)}^{2}\\)<\/td>\n66. \\(\\left(5x+y\\right)\\left(x-5y\\right)\\)<\/td>\n<\/tr>\n
67. \\(\\left({a}^{2}+4b\\right)\\left(4a-{b}^{2}\\right)\\)<\/td>\n68. \\(\\left({p}^{5}+{q}^{5}\\right)\\left({p}^{5}-{q}^{5}\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Divide a Polynomial by a Monomial<\/h2>\n

In the following exercises, divide each polynomial by the monomial.<\/p>\n\n\n\n\n\n\n\n
69. \\(\\left(35{x}^{2}-75x\\right)\\div5x\\)<\/td>\n70. \\(\\frac{42{z}^{2}-18z}{6}\\)<\/td>\n<\/tr>\n
71. \\(\\frac{550{p}^{6}-300{p}^{4}}{10{p}^{3}}\\)<\/td>\n72. \\(\\frac{81{n}^{4}+105{n}^{2}}{-3}\\)<\/td>\n<\/tr>\n
73. \\(\\frac{96{a}^{5}{b}^{2}-48{a}^{4}{b}^{3}-56{a}^{2}{b}^{4}}{8a{b}^{2}}\\)<\/td>\n74. \\(\\left(63x{y}^{3}+56{x}^{2}{y}^{4}\\right)\\div\\left(7xy\\right)\\)<\/td>\n<\/tr>\n
75. \\(\\frac{105{y}^{5}+50{y}^{3}-5y}{5{y}^{3}}\\)<\/td>\n76. \\(\\frac{57{m}^{2}-12m+1}{-3m}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Review Exercise Answers<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
1. a) binomial b) monomial c) trinomial d) trinomial e) other polynomial<\/td>\n3. a) 3 b) 4 c) 2 d) 4 e) 0<\/td>\n5. \\(5k\\)<\/td>\n<\/tr>\n
7. \\(-27c\\)<\/td>\n9. \\(7{n}^{2}\\)<\/td>\n11. \\(\\text{13a}+b\\)<\/td>\n<\/tr>\n
13. \\(13{p}^{2}-5p-1\\)<\/td>\n15. \\(7{y}^{2}-9y+4\\)<\/td>\n17. \\(5{a}^{3}+{a}^{2}+6a+2\\)<\/td>\n<\/tr>\n
19. a) \\(-26\\) b) 10 c) 22<\/td>\n21. 12,000<\/td>\n23. \\(144{n}^{8}\\)<\/td>\n<\/tr>\n
25. \\(15{a}^{2}{b}^{5}\\)<\/td>\n27. \\(-4y-52\\)<\/td>\n29. \\({p}^{2}+3p\\)<\/td>\n<\/tr>\n
31. \\(-12{u}^{2}-42u\\)<\/td>\n33. \\(3{q}^{4}-21{q}^{3}+18{q}^{2}\\)<\/td>\n35. \\(11b-44\\)<\/td>\n<\/tr>\n
37.\n\na) \\(12{y}^{2}-44y+35\\)\n\nb) \\(12{y}^{2}-44y+35\\)\n\nc) \\(12{y}^{2}-44y+35\\)<\/td>\n39. \\({y}^{2}-12y+32\\)<\/td>\n41. \\({q}^{2}+13q-48\\)<\/td>\n<\/tr>\n
43. \\({u}^{4}+{u}^{2}-30\\)<\/td>\n45. \\(16{m}^{2}{n}^{2}-2mn-3\\)<\/td>\n47.\n\na) \\(18{x}^{3}-21{x}^{2}-34x+40\\)\n\nb) \\(18{x}^{3}-21{x}^{2}-34x+40\\)<\/td>\n<\/tr>\n
49. \\(7{m}^{3}-69{m}^{2}-31m-3\\)<\/td>\n51. \\({q}^{2}-30q+225\\)<\/td>\n53. \\(64{u}^{2}+16u+1\\)<\/td>\n<\/tr>\n
55. \\(16{a}^{2}-24ab+9{b}^{2}\\)<\/td>\n57. \\({y}^{2}-\\frac{4}{25}\\)<\/td>\n59. \\(36-{r}^{2}\\)<\/td>\n<\/tr>\n
61. \\(25{p}^{8}-16{q}^{6}\\)<\/td>\n63. \\(36{a}^{2}-121\\)<\/td>\n65. \\({c}^{8}+18{c}^{4}d+81{d}^{2}\\)<\/td>\n<\/tr>\n
67. \\(4{a}^{3}+3{a}^{2}b-4{b}^{3}\\)<\/td>\n69. \\(7x-15\\)<\/td>\n71. \\(55{p}^{3}-30p\\)<\/td>\n<\/tr>\n
73. \\(12{a}^{4}-6{a}^{3}b-7a{b}^{2}\\)<\/td>\n75. \\(21{y}^{2}+10-\\frac{1}{{y}^{2}}\\)<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Chapter Practice Test<\/h1>\n\n\n\n\n\n\n\n\n\n\n
\n

In the following exercises, simplify each expression.<\/p>\n\n

\n
\n

1. \\(\\left(12{a}^{2}-7a+4\\right)+\\left(3{a}^{2}+8a-10\\right)\\)<\/p>\n\n<\/div>\n<\/div><\/td>\n

2. For the polynomial \\(10{x}^{4}+9{y}^{2}-1\\)\na) Is it a monomial, binomial, or trinomial?\nb) What is its degree?<\/td>\n<\/tr>\n
3.\\(\\left(9{p}^{2}-5p+1\\right)-\\left(2{p}^{2}-6\\right)\\)<\/td>\n4. \\(\\left(-9{r}^{4}{s}^{5}\\right)\\left(4r{s}^{7}\\right)\\)<\/td>\n<\/tr>\n
5.\\(\\left(v-9\\right)\\left(9v-5\\right)\\)<\/td>\n6.\\(\\left(m+6\\right)\\left(m+12\\right)\\)<\/td>\n<\/tr>\n
7. \\(\\left(n-6\\right)\\left({n}^{2}-5n+4\\right)\\)<\/td>\n8.\\(\\left(4c-11\\right)\\left(3c-8\\right)\\)<\/td>\n<\/tr>\n
9. \\(\\left(7p-5\\right)\\left(7p+5\\right)\\)<\/td>\n10. \\(\\left(2x-15y\\right)\\left(5x+7y\\right)\\)<\/td>\n<\/tr>\n
11.\\({\\left(9v-2\\right)}^{2}\\)<\/td>\n12. \\(\\frac{12{x}^{3}+42{x}^{2}-6x}{2x}\\)<\/td>\n<\/tr>\n
13. \\(\\frac{64{x}^{3}-x}{4x}\\)<\/td>\n14. \\(\\frac{70x{y}^{4}+95{x}^{3}y}{5xy}\\)<\/td>\n<\/tr>\n
15. \\(\\frac{{y}^{2}-5y-18}{y}\\)<\/td>\n16. A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial \\(-16{t}^{2}+1000\\) gives the height of the package \\(t\\) seconds after it was dropped. Find the height when \\(t=6\\) seconds.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Practice Test Answers<\/h1>\n\n\n\n\n\n\n\n\n
1. \\(15{a}^{2}+a-6\\)<\/td>\n2. a) Trinomial, b) 4<\/td>\n3. \\(7{p}^{2}-5p+7\\)<\/td>\n<\/tr>\n
4. \\(-36{r}^{5}{s}^{12}\\)<\/td>\n5. \\(9{v}^{2}-86v+45\\)<\/td>\n6. \\({m}^{2}+18m+72\\)<\/td>\n<\/tr>\n
7. \\({n}^{3}-11{n}^{2}+34n-24\\)<\/td>\n8. \\(12{c}^{2}-65c+88\\)<\/td>\n9. \\(49{p}^{2}-25\\)<\/td>\n<\/tr>\n
10. \\(10{x}^{2}-61xy-105{y}^{2}\\)<\/td>\n11. \\(81{v}^{2}-36v+4\\)<\/td>\n12. \\(6{x}^{2}+21x-3\\)<\/td>\n<\/tr>\n
13. \\(16{x}^{2}-\\frac{1}{4}\\)<\/td>\n14. \\(14 y^{3}+19{x}^{2}\\)<\/td>\n15. \\(y -5-\\)\\(\\frac{18}{y}\\)<\/td>\n<\/tr>\n
16. 424 feet<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","rendered":"

<\/h3>\n

Review Exercises<\/h1>\n

Identify Polynomials, Monomials, Binomials and Trinomials<\/h2>\n

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.<\/p>\n\n\n\n
1.<\/p>\n

a) \"{a}^{2}-{b}^{2}\"
\nb) \"24{d}^{3}\"
\nc) \"{x}^{2}+8x-10\"
\nd) \"{m}^{2}{n}^{2}-2mn+6\"
\ne) \"7{y}^{3}+{y}^{2}-2y-4\"<\/td>\n

2.<\/p>\n

a) \"11{c}^{4}-23{c}^{2}+1\"
\nb) \"9{p}^{3}+6{p}^{2}-p-5\"
\nc) \"\frac{3}{7}x+\frac{5}{14}\"
\nd) 10
\ne) \"2y-12\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Determine the Degree of Polynomials<\/h2>\n

In the following exercises, determine the degree of each polynomial.<\/strong><\/p>\n\n\n\n
3.<\/p>\n

a) \"5{p}^{3}-8{p}^{2}+10p-4\"<\/p>\n

b) \"-20{q}^{4}\"<\/p>\n

c) \"{x}^{2}+6x+12\"<\/p>\n

d) \"23{r}^{2}{s}^{2}-4rs+5\"<\/p>\n

e)\u00a0 100<\/td>\n

4.<\/p>\n

a) \"3{x}^{2}+9x+10\"<\/p>\n

b) \"14{a}^{2}bc\"<\/p>\n

c) \"6y+1\"<\/p>\n

d) \"{n}^{3}-4{n}^{2}+2n-8\"<\/p>\n

e) \"-19\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Add and Subtract Monomials<\/h2>\n

In the following exercises, add or subtract the monomials.<\/p>\n\n\n\n\n\n\n
5. \"-14k+19k\"<\/td>\n6. \"{\phantom{\rule{0.2em}{0ex}}\text{5y}}^{\text{3}}+8{y}^{3}\"<\/td>\n<\/tr>\n
7. \"-9c-18c\"<\/td>\n8. \"12q-\left(-6q\right)\"<\/td>\n<\/tr>\n
9. \"3{m}^{2}+7{n}^{2}-3{m}^{2}\"<\/td>\n10. \"\text{12x}-4y-9x\"<\/td>\n<\/tr>\n
11. \"\text{13a}+b\"<\/td>\n12. \"6{x}^{2}y-4x+8x{y}^{2}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Add and Subtract Polynomials<\/h2>\n

In the following exercises, add or subtract the polynomials.<\/p>\n\n\n\n\n\n
13. \"\left(9{p}^{2}-5p+3\right)+\left(4{p}^{2}-4\right)\"<\/td>\n14. \"\left(5{x}^{2}+12x+1\right)+\left(6{x}^{2}-8x+3\right)\"<\/td>\n<\/tr>\n
15. \"\left(7{y}^{2}-8y\right)-\left(y-4\right)\"<\/td>\n16. \"\left(10{m}^{2}-8m-1\right)-\left(5{m}^{2}+m-2\right)\"<\/td>\n<\/tr>\n
17. Find the sum of \"\left({a}^{2}+6a+9\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(5{a}^{3}-7\right)\"<\/td>\n18. Subtract
\n\"\left(3{s}^{2}+10\right)\phantom{\rule{0.2em}{0ex}}\text{from}\phantom{\rule{0.2em}{0ex}}\left(15{s}^{2}-2s+8\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Evaluate a Polynomial for a Given Value of the Variable<\/h2>\n

In the following exercises, evaluate each polynomial for the given value.<\/p>\n\n\n\n\n
\n

19. Evaluate \"10-12x\" when:<\/p>\n

a) \"x=3\"<\/p>\n

b) \"x=0\"<\/p>\n

c) \"x=-1\"<\/td>\n

\n

20. Evaluate \"3{y}^{2}-y+1\" when:<\/p>\n

a) \"y=5\"<\/p>\n

b) \"y=-1\"<\/p>\n

c) \"y=0\"<\/td>\n<\/tr>\n

21. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p<\/em> dollars each is given by the polynomial \"-4{p}^{2}+460p\". Find the revenue received when \"p=75\" dollars.<\/td>\n22. Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial \"-16{t}^{2}+200\" gives the height of a stone \"t\" seconds after it is dropped from the cliff. Find the height after \"t=3\" seconds.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Multiply Monomials<\/h2>\n

In the following exercises, multiply the monomials.<\/p>\n\n\n\n\n
23. \"\left(-9{n}^{7}\right)\left(-16n\right)\"<\/td>\n24. \"\left(-15{x}^{2}\right)\left(6{x}^{4}\right)\"<\/td>\n<\/tr>\n
25. \"\left(\frac{5}{9}a{b}^{2}\right)\left(27a{b}^{3}\right)\"<\/td>\n26. \"\left(7{p}^{5}{q}^{3}\right)\left(8p{q}^{9}\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Multiply a Polynomial by a Monomial<\/h2>\n

In the following exercises, multiply.<\/p>\n\n\n\n\n\n\n\n
27. \"-4\left(y+13\right)\"<\/td>\n28. \"7\left(a+9\right)\"<\/td>\n<\/tr>\n
29. \"p\left(p+3\right)\"<\/td>\n30. \"-5\left(r-2\right)\"<\/td>\n<\/tr>\n
31. \"-6u\left(2u+7\right)\"<\/td>\n32. \"-m\left(m+15\right)\"<\/td>\n<\/tr>\n
33. \"3{q}^{2}\left({q}^{2}-7q+6\right)\" 3<\/td>\n34. \"9\left({b}^{2}+6b+8\right)\"<\/td>\n<\/tr>\n
35. \"\left(b-4\right)\cdot 11\"<\/td>\n36. \"\left(5z-1\right)z\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Multiply a Binomial by a Binomial<\/h2>\n

In the following exercises, multiply the binomials using: a) the Distributive Property, b) the FOIL method, c) the Vertical Method.<\/p>\n\n\n\n
37. \"\left(6y-7\right)\left(2y-5\right)\"<\/td>\n38. \"\left(x-4\right)\left(x+10\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

In the following exercises, multiply the binomials. Use any method.<\/p>\n\n\n\n\n\n\n
39. \"\left(y-4\right)\left(y-8\right)\"<\/td>\n40. \"\left(x+3\right)\left(x+9\right)\"<\/td>\n<\/tr>\n
41. \"\left(q+16\right)\left(q-3\right)\"<\/td>\n42. \"\left(p-7\right)\left(p+4\right)\"<\/td>\n<\/tr>\n
43. \"\left({u}^{2}+6\right)\left({u}^{2}-5\right)\"<\/td>\n44. \"\left(5m-8\right)\left(12m+1\right)\"<\/td>\n<\/tr>\n
45. \"\left(8mn+3\right)\left(2mn-1\right)\"<\/td>\n46. \"\left(9x-y\right)\left(6x-5\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
\n
<\/div>\n<\/div>\n

Multiply a Trinomial by a Binomial<\/h2>\n

In the following exercises, multiply using a) the Distributive Property, b) the Vertical Method.<\/p>\n\n\n\n
47. \"\left(3x-4\right)\left(6{x}^{2}+x-10\right)\"<\/td>\n48. \"\left(n+1\right)\left({n}^{2}+5n-2\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

In the following exercises, multiply. Use either method.<\/p>\n\n\n\n
49. \"\left(7m+1\right)\left({m}^{2}-10m-3\right)\"<\/td>\n50. \"\left(y-2\right)\left({y}^{2}-8y+9\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Square a Binomial Using the Binomial Squares Pattern<\/h2>\n

In the following exercises, square each binomial using the Binomial Squares Pattern.<\/p>\n\n\n\n\n\n
51. \"{\left(q-15\right)}^{2}\"<\/td>\n52. \"{\left(c+11\right)}^{2}\"<\/td>\n<\/tr>\n
53. \"{\left(8u+1\right)}^{2}\"<\/td>\n54. \"{\left(x+\frac{1}{3}\right)}^{2}\"<\/td>\n<\/tr>\n
55. \"{\left(4a-3b\right)}^{2}\"<\/td>\n56. \"{\left(3{n}^{3}-2\right)}^{2}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Multiply Conjugates Using the Product of Conjugates Pattern<\/h2>\n

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.<\/p>\n\n\n\n\n\n
57. \"\left(y+\frac{2}{5}\right)\left(y-\frac{2}{5}\right)\"<\/td>\n58. \"\left(s-7\right)\left(s+7\right)\"<\/td>\n<\/tr>\n
59. \"\left(6-r\right)\left(6+r\right)\"<\/td>\n60. \"\left(12c+13\right)\left(12c-13\right)\"<\/td>\n<\/tr>\n
61. \"\left(5{p}^{4}-4{q}^{3}\right)\left(5{p}^{4}+4{q}^{3}\right)\"<\/td>\n62. \"\left(u+\frac{3}{4}v\right)\left(u-\frac{3}{4}v\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Recognize and Use the Appropriate Special Product Pattern<\/h2>\n

In the following exercises, find each product.<\/p>\n\n\n\n\n\n
63. \"\left(6a+11\right)\left(6a-11\right)\"<\/td>\n64. \"{\left(3m+10\right)}^{2}\"<\/td>\n<\/tr>\n
65. \"{\left({c}^{4}+9d\right)}^{2}\"<\/td>\n66. \"\left(5x+y\right)\left(x-5y\right)\"<\/td>\n<\/tr>\n
67. \"\left({a}^{2}+4b\right)\left(4a-{b}^{2}\right)\"<\/td>\n68. \"\left({p}^{5}+{q}^{5}\right)\left({p}^{5}-{q}^{5}\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Divide a Polynomial by a Monomial<\/h2>\n

In the following exercises, divide each polynomial by the monomial.<\/p>\n\n\n\n\n\n\n
69. \"\left(35{x}^{2}-75x\right)\div5x\"<\/td>\n70. \"\frac{42{z}^{2}-18z}{6}\"<\/td>\n<\/tr>\n
71. \"\frac{550{p}^{6}-300{p}^{4}}{10{p}^{3}}\"<\/td>\n72. \"\frac{81{n}^{4}+105{n}^{2}}{-3}\"<\/td>\n<\/tr>\n
73. \"\frac{96{a}^{5}{b}^{2}-48{a}^{4}{b}^{3}-56{a}^{2}{b}^{4}}{8a{b}^{2}}\"<\/td>\n74. \"\left(63x{y}^{3}+56{x}^{2}{y}^{4}\right)\div\left(7xy\right)\"<\/td>\n<\/tr>\n
75. \"\frac{105{y}^{5}+50{y}^{3}-5y}{5{y}^{3}}\"<\/td>\n76. \"\frac{57{m}^{2}-12m+1}{-3m}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Review Exercise Answers<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
1. a) binomial b) monomial c) trinomial d) trinomial e) other polynomial<\/td>\n3. a) 3 b) 4 c) 2 d) 4 e) 0<\/td>\n5. \"5k\"<\/td>\n<\/tr>\n
7. \"-27c\"<\/td>\n9. \"7{n}^{2}\"<\/td>\n11. \"\text{13a}+b\"<\/td>\n<\/tr>\n
13. \"13{p}^{2}-5p-1\"<\/td>\n15. \"7{y}^{2}-9y+4\"<\/td>\n17. \"5{a}^{3}+{a}^{2}+6a+2\"<\/td>\n<\/tr>\n
19. a) \"-26\" b) 10 c) 22<\/td>\n21. 12,000<\/td>\n23. \"144{n}^{8}\"<\/td>\n<\/tr>\n
25. \"15{a}^{2}{b}^{5}\"<\/td>\n27. \"-4y-52\"<\/td>\n29. \"{p}^{2}+3p\"<\/td>\n<\/tr>\n
31. \"-12{u}^{2}-42u\"<\/td>\n33. \"3{q}^{4}-21{q}^{3}+18{q}^{2}\"<\/td>\n35. \"11b-44\"<\/td>\n<\/tr>\n
37.<\/p>\n

a) \"12{y}^{2}-44y+35\"<\/p>\n

b) \"12{y}^{2}-44y+35\"<\/p>\n

c) \"12{y}^{2}-44y+35\"<\/td>\n

39. \"{y}^{2}-12y+32\"<\/td>\n41. \"{q}^{2}+13q-48\"<\/td>\n<\/tr>\n
43. \"{u}^{4}+{u}^{2}-30\"<\/td>\n45. \"16{m}^{2}{n}^{2}-2mn-3\"<\/td>\n47.<\/p>\n

a) \"18{x}^{3}-21{x}^{2}-34x+40\"<\/p>\n

b) \"18{x}^{3}-21{x}^{2}-34x+40\"<\/td>\n<\/tr>\n

49. \"7{m}^{3}-69{m}^{2}-31m-3\"<\/td>\n51. \"{q}^{2}-30q+225\"<\/td>\n53. \"64{u}^{2}+16u+1\"<\/td>\n<\/tr>\n
55. \"16{a}^{2}-24ab+9{b}^{2}\"<\/td>\n57. \"{y}^{2}-\frac{4}{25}\"<\/td>\n59. \"36-{r}^{2}\"<\/td>\n<\/tr>\n
61. \"25{p}^{8}-16{q}^{6}\"<\/td>\n63. \"36{a}^{2}-121\"<\/td>\n65. \"{c}^{8}+18{c}^{4}d+81{d}^{2}\"<\/td>\n<\/tr>\n
67. \"4{a}^{3}+3{a}^{2}b-4{b}^{3}\"<\/td>\n69. \"7x-15\"<\/td>\n71. \"55{p}^{3}-30p\"<\/td>\n<\/tr>\n
73. \"12{a}^{4}-6{a}^{3}b-7a{b}^{2}\"<\/td>\n75. \"21{y}^{2}+10-\frac{1}{{y}^{2}}\"<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Chapter Practice Test<\/h1>\n\n\n\n\n\n\n\n\n\n\n
\n

In the following exercises, simplify each expression.<\/p>\n

\n
\n

1. \"\left(12{a}^{2}-7a+4\right)+\left(3{a}^{2}+8a-10\right)\"<\/p>\n<\/div>\n<\/div>\n<\/td>\n

2. For the polynomial \"10{x}^{4}+9{y}^{2}-1\"
\na) Is it a monomial, binomial, or trinomial?
\nb) What is its degree?<\/td>\n<\/tr>\n
3.\"\left(9{p}^{2}-5p+1\right)-\left(2{p}^{2}-6\right)\"<\/td>\n4. \"\left(-9{r}^{4}{s}^{5}\right)\left(4r{s}^{7}\right)\"<\/td>\n<\/tr>\n
5.\"\left(v-9\right)\left(9v-5\right)\"<\/td>\n6.\"\left(m+6\right)\left(m+12\right)\"<\/td>\n<\/tr>\n
7. \"\left(n-6\right)\left({n}^{2}-5n+4\right)\"<\/td>\n8.\"\left(4c-11\right)\left(3c-8\right)\"<\/td>\n<\/tr>\n
9. \"\left(7p-5\right)\left(7p+5\right)\"<\/td>\n10. \"\left(2x-15y\right)\left(5x+7y\right)\"<\/td>\n<\/tr>\n
11.\"{\left(9v-2\right)}^{2}\"<\/td>\n12. \"\frac{12{x}^{3}+42{x}^{2}-6x}{2x}\"<\/td>\n<\/tr>\n
13. \"\frac{64{x}^{3}-x}{4x}\"<\/td>\n14. \"\frac{70x{y}^{4}+95{x}^{3}y}{5xy}\"<\/td>\n<\/tr>\n
15. \"\frac{{y}^{2}-5y-18}{y}\"<\/td>\n16. A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial \"-16{t}^{2}+1000\" gives the height of the package \"t\" seconds after it was dropped. Find the height when \"t=6\" seconds.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Practice Test Answers<\/h1>\n\n\n\n\n\n\n\n\n
1. \"15{a}^{2}+a-6\"<\/td>\n2. a) Trinomial, b) 4<\/td>\n3. \"7{p}^{2}-5p+7\"<\/td>\n<\/tr>\n
4. \"-36{r}^{5}{s}^{12}\"<\/td>\n5. \"9{v}^{2}-86v+45\"<\/td>\n6. \"{m}^{2}+18m+72\"<\/td>\n<\/tr>\n
7. \"{n}^{3}-11{n}^{2}+34n-24\"<\/td>\n8. \"12{c}^{2}-65c+88\"<\/td>\n9. \"49{p}^{2}-25\"<\/td>\n<\/tr>\n
10. \"10{x}^{2}-61xy-105{y}^{2}\"<\/td>\n11. \"81{v}^{2}-36v+4\"<\/td>\n12. \"6{x}^{2}+21x-3\"<\/td>\n<\/tr>\n
13. \"16{x}^{2}-\frac{1}{4}\"<\/td>\n14. \"14 y^{3}+19{x}^{2}\"<\/td>\n15. \"y -5-\"\"\frac{18}{y}\"<\/td>\n<\/tr>\n
16. 424 feet<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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