5.4E: Exercises
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Practice Makes Perfect
Multiply Monomials
In the following exercises, multiply the monomials.
1. ⓐ (6y^7)(−3y^4) ⓑ (\frac{4}{7}rs^2)(\frac{1}{4}rs^3)
2. ⓐ (−10x^5)(−3x^3) ⓑ (58x^3y)(24x^5y)
- Answer
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ⓐ30x^8 ⓑ 15x^8y^2
3. ⓐ (−8u^6)(−9u) ⓑ (\frac{2}{3}x^2y)(\frac{3}{4}xy^2)
4. ⓐ (−6c^4)(−12c) ⓑ (\frac{3}{5}m^3n^2)(\frac{5}{9}m^2n^3)
- Answer
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ⓐ 72c^5 ⓑ \frac{1}{3}m^5n^5
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
5. ⓐ−8x(x^2+2x−15) ⓑ 5pq^3(p^2−2pq+6q^2)
6. ⓐ −5t(t^2+3t−18) ⓑ 9r^3s(r^2−3rs+5s^2)
- Answer
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ⓐ −5t^3−15t^2+90t
ⓑ 9sr^5−27s^2r^4+45s^3r^3
7. ⓐ −8y(y^2+2y−15) ⓑ −4y^2z^2(3y^2+12yz−z^2)
8. ⓐ −5m(m^2+3m−18) ⓑ −3x^2y^2(7x^2+10xy−y^2)
- Answer
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ⓐ −5m^3−15m^2+90m
ⓑ −21x^4y^2−30x^3y^3+3x^2y^4
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using ⓐ the Distributive Property; ⓑ the FOIL method; ⓒ the Vertical Method.
9. (w+5)(w+7)
10. (y+9)(y+3)
- Answer
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y^2+12y+27
11. (4p+11)(5p−4)
12. (7q+4)(3q−8)
- Answer
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21q^2−44q−32
In the following exercises, multiply the binomials. Use any method.
13. (x+8)(x+3)
14. (y−6)(y−2)
- Answer
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y^2−8y+12
15. (2t−9)(10t+1)
16. (6p+5)(p+1)
- Answer
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6p^2+11p+5
17. (q−5)(q+8)
18. (m+11)(m−4)
- Answer
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m^2+7m−44
19. (7m+1)(m−3)
20. (3r−8)(11r+1)
- Answer
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33r^2−85r−8
21. (x^2+3)(x+2)
22. (y^2−4)(y+3)
- Answer
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y^3+3y^2−4y−12
23. (5ab−1)(2ab+3)
24. (2xy+3)(3xy+2)
- Answer
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6x^2y^2+13xy+6
25. (x^2+8)(x^2−5)
26. (y^2−7)(y^2−4)
- Answer
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y^4−11y^2+28
27. (6pq−3)(4pq−5)
28. (3rs−7)(3rs−4)
- Answer
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9r^2s^2−33rs+28
Multiply a Polynomial by a Polynomial
In the following exercises, multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.
29. (x+5)(x^2+4x+3)
30. (u+4)(u^2+3u+2)
- Answer
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u^3+7u^2+14u+8
31. (y+8)(4y^2+y−7)
32. (a+10)(3a^2+a−5)
- Answer
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3a^3+31a^2+5a−50
33. (y^2−3y+8)(4y^2+y−7)
34. (2a^2−5a+10)(3a^2+a−5)
- Answer
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6a^4−13a^3+15a^2+35a−50
Multiply Special Products
In the following exercises, multiply. Use either method.
35. (w−7)(w^2−9w+10)
36. (p−4)(p^2−6p+9)
- Answer
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p^3−10p^2+33p−36
37. (3q+1)(q^2−4q−5)
38. (6r+1)(r^2−7r−9)
- Answer
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6r^3−41r^2−61r−9
In the following exercises, square each binomial using the Binomial Squares Pattern.
39. (w+4)^2
40. (q+12)^2
- Answer
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q^2+24q+144
41. (3x−y)^2
42. (2y−3z)^2
- Answer
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4y^2−12yz+9z^2
43. (y+\frac{1}{4})^2
44. (x+\frac{2}{3})^2
- Answer
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x^2+\frac{4}{3}x+\frac{4}{9}
45. (\frac{1}{5}x−\frac{1}{7}y)^2
46. (\frac{1}{8}x−\frac{1}{9}y)^2
- Answer
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\frac{1}{64}x^2−\frac{1}{36}xy+\frac{1}{81}y^2
47. (3x^2+2)^2
48. (5u^2+9)^2
- Answer
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25u^4+90u^2+81
49. (4y3−2)2
50. (8p3−3)2
- Answer
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64p^6−48p^3+9
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
51. (5k+6)(5k−6)
52. (8j+4)(8j−4)
- Answer
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64j^2−16
53. (11k+4)(11k−4)
54. (9c+5)(9c−5)
- Answer
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81c^2−25
55. (9c−2d)(9c+2d)
56. (7w+10x)(7w−10x)
- Answer
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49w^2−100x^2
57. (m+\frac{2}{3}n)(m−\frac{2}{3}n)
58. (p+\frac{4}{5}q)(p−\frac{4}{5}q)
- Answer
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p^2−\frac{16}{25}q^2
59. (ab−4)(ab+4)
60. (xy−9)(xy+9)
- Answer
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x^2y^2−81
61. (12p^3−11q^2)(12p^3+11q^2)
62. (15m^2−8n^4)(15m^2+8n^4)
- Answer
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225m^4−64n^8
In the following exercises, find each product.
63. (p−3)(p+3)
64. (t−9)^2
- Answer
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t^2−18t+81
65. (m+n)^2
66. (2x+y)(x−2y)
- Answer
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2x^2−3xy−2y^2
67. (2r+12)^2
68. (3p+8)(3p−8)
- Answer
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9p^2−64
69. (7a+b)(a−7b)
70. (k−6)^2
- Answer
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k^2−12k+36
71. (a^5−7b)^2
72. (x^2+8y)(8x−y^2)
- Answer
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8x^3−x^2y^2+64xy−8y^3
73. (r^6+s^6)(r^6−s^6)
74. (y^4+2z)^2
- Answer
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y^8+4y^4z+4z^2
75. (x^5+y^5)(x^5−y^5)
76. (m^3−8n)^2
- Answer
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m^6−16m^3n+64n^2
77. (9p+8q)^2
78. (r^2−s^3)(r^3+s^2)
- Answer
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r^5+r^2s^2−r^3s^3−s^5
Mixed Practice
79. (10y−6)+(4y−7)
80. (15p−4)+(3p−5)
- Answer
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18p−9
81. (x^2−4x−34)−(x^2+7x−6)
82. (j^2−8j−27)−(j^2+2j−12)
- Answer
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−10j−15
83. (\frac{1}{5}f^8)(20f^3)
84. (\frac{1}{4}d^5)(36d^2)
- Answer
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9d^7
85. (4a^3b)(9a^2b^6)
86. (6m^4n^3)(7mn^5)
- Answer
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72m^5n^8
87. −5m(m^2+3m−18)
88. 5q^3(q^2−2q+6)
- Answer
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5q^5−10q^4+30q^3
89. (s−7)(s+9)
90. (y^2−2y)(y+1)
- Answer
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y^3−y^2−2y
91. (5x−y)(x−4)
92. (6k−1)(k^2+2k−4)
- Answer
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6k^3−11k^2−26k+4
93. (3x−11y)(3x−11y)
94. (11−b)(11+b)
- Answer
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121−b^2
95. (rs−\frac{2}{7})(rs+\frac{2}{7})
96. (2x^2−3y^4)(2x^2+3y^4)
- Answer
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4x^4−9y^8
97. (m−15)^2
98. (3d+1)^2
- Answer
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9d^2+6d+1
99. (4a+10)^2
100. (3z+15)^2
- Answer
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9z^2−\frac{6}{5}z+\frac{1}{25}
Multiply Polynomial Functions
101. For functions f(x)=x+2 and g(x)=3x^2−2x+4, find ⓐ (f·g)(x) ⓑ (f·g)(−1)
102. For functions f(x)=x−1 and g(x)=4x^2+3x−5, find ⓐ (f·g)(x) ⓑ (f·g)(−2)
- Answer
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ⓐ (f·g)(x)=4x^3−x^2−8x+5
ⓑ (f·g)(−2)=−15
103. For functions f(x)=2x−7 and g(x)=2x+7, find ⓐ (f·g)(x) ⓑ (f·g)(−3)
104. For functions f(x)=7x−8 and g(x)=7x+8, find ⓐ (f·g)(x) ⓑ (f·g)(−2)
- Answer
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ⓐ (f·g)(x)=49x^2−64
ⓑ (f·g)(−2)=187
105. For functions f(x)=x^2−5x+2 and g(x)=x^2−3x−1, find ⓐ (f·g)(x) ⓑ (f·g)(−1)
106. For functions f(x)=x^2+4x−3 and g(x)=x^2+2x+4, find ⓐ (f·g)(x) ⓑ (f·g)(1)
- Answer
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ⓐ (f·g)(x)=x^4+6x^3+9x^2+10x−12 ⓑ (f·g)(1)=14
Writing Exercises
107. Which method do you prefer to use when multiplying two binomials: the Distributive Property or the FOIL method? Why? Which method do you prefer to use when multiplying a polynomial by a polynomial: the Distributive Property or the Vertical Method? Why?
108. Multiply the following:
(x+2)(x−2)
(y+7)(y−7)
(w+5)(w−5)
Explain the pattern that you see in your answers.
- Answer
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Answers will vary.
109. Multiply the following:
(p+3)(p+3)
(q+6)(q+6)
(r+1)(r+1)
Explain the pattern that you see in your answers.
110. Why does (a+b)^2 result in a trinomial, but (a−b)(a+b) result in a binomial?
- Answer
-
Answers will vary.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?