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6.8 Chapter Review

Review Exercises

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

1.

a) {a}^{2}-{b}^{2}
b) 24{d}^{3}
c) {x}^{2}+8x-10
d) {m}^{2}{n}^{2}-2mn+6
e) 7{y}^{3}+{y}^{2}-2y-4

 2.

a) 11{c}^{4}-23{c}^{2}+1
b) 9{p}^{3}+6{p}^{2}-p-5
c) \frac{3}{7}x+\frac{5}{14}
d) 10
e) 2y-12

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

3.

a) 5{p}^{3}-8{p}^{2}+10p-4

b) -20{q}^{4}

c) {x}^{2}+6x+12

d) 23{r}^{2}{s}^{2}-4rs+5

e)  100

 4.

a) 3{x}^{2}+9x+10

b) 14{a}^{2}bc

c) 6y+1

d) {n}^{3}-4{n}^{2}+2n-8

e) -19

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

5. -14k+19k 6. {\phantom{\rule{0.2em}{0ex}}\text{5y}}^{\text{3}}+8{y}^{3}
7. -9c-18c 8. 12q-\left(-6q\right)
9. 3{m}^{2}+7{n}^{2}-3{m}^{2} 10. \text{12x}-4y-9x
11. \text{13a}+b 12. 6{x}^{2}y-4x+8x{y}^{2}

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

13. \left(9{p}^{2}-5p+3\right)+\left(4{p}^{2}-4\right) 14. \left(5{x}^{2}+12x+1\right)+\left(6{x}^{2}-8x+3\right)
15. \left(7{y}^{2}-8y\right)-\left(y-4\right) 16. \left(10{m}^{2}-8m-1\right)-\left(5{m}^{2}+m-2\right)
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) 18. Subtract
\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)
Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

19. Evaluate 10-12x when:

a) x=3

b) x=0

c) x=-1

20. Evaluate 3{y}^{2}-y+1 when:

a) y=5

b) y=-1

c) y=0
21. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial -4{p}^{2}+460p. Find the revenue received when p=75 dollars. 22. 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.
Multiply Monomials

In the following exercises, multiply the monomials.

23. \left(-9{n}^{7}\right)\left(-16n\right) 24. \left(-15{x}^{2}\right)\left(6{x}^{4}\right)
25. \left(\frac{5}{9}a{b}^{2}\right)\left(27a{b}^{3}\right) 26. \left(7{p}^{5}{q}^{3}\right)\left(8p{q}^{9}\right)

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

27. -4\left(y+13\right) 28. 7\left(a+9\right)
29. p\left(p+3\right) 30. -5\left(r-2\right)
31. -6u\left(2u+7\right) 32. -m\left(m+15\right)
33. 3{q}^{2}\left({q}^{2}-7q+6\right) 34. 9\left({b}^{2}+6b+8\right)
35. \left(b-4\right)\cdot 11 36. \left(5z-1\right)z

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using: a) the Distributive Property, b) the FOIL method, c) the Vertical Method.

37. \left(6y-7\right)\left(2y-5\right) 38. \left(x-4\right)\left(x+10\right)

In the following exercises, multiply the binomials. Use any method.

39. \left(y-4\right)\left(y-8\right) 40. \left(x+3\right)\left(x+9\right)
41. \left(q+16\right)\left(q-3\right) 42. \left(p-7\right)\left(p+4\right)
43. \left({u}^{2}+6\right)\left({u}^{2}-5\right) 44. \left(5m-8\right)\left(12m+1\right)
45. \left(8mn+3\right)\left(2mn-1\right) 46. \left(9x-y\right)\left(6x-5\right)
Multiply a Trinomial by a Binomial

In the following exercises, multiply using a) the Distributive Property, b) the Vertical Method.

47. \left(3x-4\right)\left(6{x}^{2}+x-10\right) 48. \left(n+1\right)\left({n}^{2}+5n-2\right)

In the following exercises, multiply. Use either method.

49. \left(7m+1\right)\left({m}^{2}-10m-3\right) 50. \left(y-2\right)\left({y}^{2}-8y+9\right)

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

51. {\left(q-15\right)}^{2} 52. {\left(c+11\right)}^{2}
53. {\left(8u+1\right)}^{2} 54. {\left(x+\frac{1}{3}\right)}^{2}
55. {\left(4a-3b\right)}^{2} 56. {\left(3{n}^{3}-2\right)}^{2}
Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

57. \left(y+\frac{2}{5}\right)\left(y-\frac{2}{5}\right) 58. \left(s-7\right)\left(s+7\right)
59. \left(6-r\right)\left(6+r\right) 60. \left(12c+13\right)\left(12c-13\right)
61. \left(5{p}^{4}-4{q}^{3}\right)\left(5{p}^{4}+4{q}^{3}\right) 62. \left(u+\frac{3}{4}v\right)\left(u-\frac{3}{4}v\right)
Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

63. \left(6a+11\right)\left(6a-11\right) 64. {\left(3m+10\right)}^{2}
65. {\left({c}^{4}+9d\right)}^{2} 66. \left(5x+y\right)\left(x-5y\right)
67. \left({a}^{2}+4b\right)\left(4a-{b}^{2}\right) 68. \left({p}^{5}+{q}^{5}\right)\left({p}^{5}-{q}^{5}\right)

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

69. \left(35{x}^{2}-75x\right)\div5x 70. \frac{42{z}^{2}-18z}{6}
71. \frac{550{p}^{6}-300{p}^{4}}{10{p}^{3}} 72. \frac{81{n}^{4}+105{n}^{2}}{-3}
73. \frac{96{a}^{5}{b}^{2}-48{a}^{4}{b}^{3}-56{a}^{2}{b}^{4}}{8a{b}^{2}} 74. \left(63x{y}^{3}+56{x}^{2}{y}^{4}\right)\div\left(7xy\right)
75. \frac{105{y}^{5}+50{y}^{3}-5y}{5{y}^{3}} 76. \frac{57{m}^{2}-12m+1}{-3m}

Greatest Common Factor of two or more expressions

In the following exercises, find the greatest common factor.

77. 56, 24 78. 3 , 18
79. 58r , 38r^2 80. 54r^5m^5 , 14rm^4
81. 32m^7y^9, 31m^5y^2, 34m^8y^7 82. 14t^3p^7, 27t^5p, 3t^2p^2

Factor the Greatest Common Factor from a polynomial

In the following exercises, factor the greatest common factor from each polynomial.

83. 3z+9 84. 5m+5
85. -26t-44 86. -12t-33
87. 6t^3-12t^2+15t 88. 5p^2-15p-5
89. 6m(m+5)-7(m+5) 90. z(z-9)-8(z-9)

Factor by Grouping

In the following exercises, factor by grouping.

91. mn-4n+6m-24 92. at+10t-7a-70
93. xy-10x+8y-80 94. a^2-7a+5a-35

Factor Trinomials of the Form x^2+bx+c

In the following exercises, factor each trinomial.

95. -11-10y+y^2 96. t^2+3t+8
97. x^2-9x+42 98. w^2+4w+16
99. 7s+s^2+6 100. 5x+x^2-14

Factor Trinomials of the Form x^2+bxy+cy^2

In the following exercises, factor each trinomial.

101. m^2+6mn+5n^2 102. b^2+12by+27y^2
103. v^2-10vx-24x^2 104. x^2+10xb+16b^2
105. p^2-8pq-65q^2 106. t^2-3ts+2s^2

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

107. \frac{6x^2-4}{2} 108. \frac{21y^2-7}{7}
109. \frac{-30x^4-6}{-6} 110. \frac{6c^5-6c^4}{6c^3}
111. (4y^3-10y^5) \div 2y^2 112. (14r^4y^3-28r^4y^4) \div 7r^3y^2
113. \frac{66{x}^{3}{y}^{2}-110{x}^{2}{y}^{3}-44{x}^{4}{y}^{3}}{11{x}^{2}{y}^{2}} 114. \frac{4{w}^{2}+2w-5}{2w}

Factor Binomials of the Form a^2x^2-b^2y^2

In the following exercises, factorize using the difference of squares.

115. {x}^2 - \dfrac{9}{16} 116. 25k^2 - 36
117. 81c^2 - 4d^2 118. {u}^{2}{v}^{2}-\dfrac{9}{25}
119. 9y^2 - (y+9)^2 120. 25 - (n+3)^2

Review Exercise Answers

1. a) binomial b) monomial c) trinomial d) trinomial e) other polynomial 3. a) 3 b) 4 c) 2 d) 4 e) 0 5. 5k
7. -27c 9. 7{n}^{2} 11. \text{13a}+b
13. 13{p}^{2}-5p-1 15. 7{y}^{2}-9y+4 17. 5{a}^{3}+{a}^{2}+6a+2
19. a) -26 b) 10 c) 22 21. 12,000 23. 144{n}^{8}
25. 15{a}^{2}{b}^{5} 27. -4y-52 29. {p}^{2}+3p
31. -12{u}^{2}-42u 33. 3{q}^{4}-21{q}^{3}+18{q}^{2} 35. 11b-44
37.

a) 12{y}^{2}-44y+35

b) 12{y}^{2}-44y+35

c) 12{y}^{2}-44y+35

 39. {y}^{2}-12y+32 41. {q}^{2}+13q-48
43. {u}^{4}+{u}^{2}-30 45. 16{m}^{2}{n}^{2}-2mn-3 47.

a) 18{x}^{3}-21{x}^{2}-34x+40

b) 18{x}^{3}-21{x}^{2}-34x+40
49. 7{m}^{3}-69{m}^{2}-31m-3 51. {q}^{2}-30q+225 53. 64{u}^{2}+16u+1
55. 16{a}^{2}-24ab+9{b}^{2} 57. {y}^{2}-\frac{4}{25} 59. 36-{r}^{2}
61. 25{p}^{8}-16{q}^{6} 63. 36{a}^{2}-121 65. {c}^{8}+18{c}^{4}d+81{d}^{2}
67. 4{a}^{3}+3{a}^{2}b-4{b}^{3} 69. 7x-15 71. 55{p}^{3}-30p
73. 12{a}^{4}-6{a}^{3}b-7a{b}^{2} 75. 21{y}^{2}+10-\frac{1}{{y}^{2}} 77. 8
79.  2r 81. m^5y^2 83. 3(z+3)
85. -2(13t+22) 87. 3t(2t^2-4t+5) 89. (6m-7)(m+5)
91. (n+6)(m-4) 93. (x+8)(y-10) 95. (y-11)(y+1)
97. Prime or not factorable. 99. (s+6)(s+1) 101. (m+5n)(m+n)
103. (v+2x)(v-12x) 105. (p-13q)(p+5q) 107. 3x^2-2
109. 5x^4+1 111. 2y-5y^3 113. 6x-10y-4x^2y
115. (x+\frac{3}{4})(x-\frac{3}{4}) 117. (9c+2d)(9c-2d) 119. (4y+9)(2y-9)

Chapter Practice Test

In the following exercises, simplify each expression.

1. \left(12{a}^{2}-7a+4\right)+\left(3{a}^{2}+8a-10\right)

 2. For the polynomial 10{x}^{4}+9{y}^{2}-1
a) Is it a monomial, binomial, or trinomial?
b) What is its degree?
3.\left(9{p}^{2}-5p+1\right)-\left(2{p}^{2}-6\right) 4. \left(-9{r}^{4}{s}^{5}\right)\left(4r{s}^{7}\right)
5.\left(v-9\right)\left(9v-5\right) 6.\left(m+6\right)\left(m+12\right)
7. \left(n-6\right)\left({n}^{2}-5n+4\right) 8.\left(4c-11\right)\left(3c-8\right)
9. \left(7p-5\right)\left(7p+5\right) 10. \left(2x-15y\right)\left(5x+7y\right)
11.{\left(9v-2\right)}^{2} 12. \frac{12{x}^{3}+42{x}^{2}-6x}{2x}
13. \frac{64{x}^{3}-x}{4x} 14. \frac{70x{y}^{4}+95{x}^{3}y}{5xy}
15. \frac{{y}^{2}-5y-18}{y} 16.  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.
17.  Divide: \frac{51z^4+42z^2}{-3z^2} 18. Divide: (42a^4b^3-28a^4b^4) \div 7a^3b^3
19. Multiply: (2x+y)^2 20. Factor: 8y^2(3y-8)-7(3y-8)
21. Factor: 25x^3+5x^2+30x+6 22. Factor: m^2-29mn-96n^2
23. Factor: x^2-9x+18 24. Factor:  0.36x^2 - 0.49 y^2
Practice Test Answers
1. 15{a}^{2}+a-6 2. a) Trinomial, b) 4 3. 7{p}^{2}-5p+7
4. -36{r}^{5}{s}^{12} 5. 9{v}^{2}-86v+45 6. {m}^{2}+18m+72
7. {n}^{3}-11{n}^{2}+34n-24 8. 12{c}^{2}-65c+88 9. 49{p}^{2}-25
10. 10{x}^{2}-61xy-105{y}^{2} 11. 81{v}^{2}-36v+4 12. 6{x}^{2}+21x-3
13. 16{x}^{2}-\frac{1}{4} 14. 14 y^{3}+19{x}^{2} 15. y -5-\frac{18}{y}
16. 424 feet 17. -17z^2-14 18. 6a-4ab
19. 4x^2+4xy+y^2 20. (8y^2-7)(3y-8) 21. (5x^2+6)(5x+1)
22. (m-32n)(m+3n) 23. (x-3)(x-6) 24. 0.01(6x+7y)(6x-7y) \text{OR} (0.6x+0.7y)(0.6x-0.7y)

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