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Mathematics LibreTexts

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

30x^815x^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

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

−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

−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

y^2+12y+27

11. (4p+11)(5p−4)

12. (7q+4)(3q−8)

Answer

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

y^2−8y+12

15. (2t−9)(10t+1)

16. (6p+5)(p+1)

Answer

6p^2+11p+5

17. (q−5)(q+8)

18. (m+11)(m−4)

Answer

m^2+7m−44

19. (7m+1)(m−3)

20. (3r−8)(11r+1)

Answer

33r^2−85r−8

21. (x^2+3)(x+2)

22. (y^2−4)(y+3)

Answer

y^3+3y^2−4y−12

23. (5ab−1)(2ab+3)

24. (2xy+3)(3xy+2)

Answer

6x^2y^2+13xy+6

25. (x^2+8)(x^2−5)

26. (y^2−7)(y^2−4)

Answer

y^4−11y^2+28

27. (6pq−3)(4pq−5)

28. (3rs−7)(3rs−4)

Answer

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

u^3+7u^2+14u+8

31. (y+8)(4y^2+y−7)

32. (a+10)(3a^2+a−5)

Answer

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

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

p^3−10p^2+33p−36

37. (3q+1)(q^2−4q−5)

38. (6r+1)(r^2−7r−9)

Answer

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

q^2+24q+144

41. (3x−y)^2

42. (2y−3z)^2

Answer

4y^2−12yz+9z^2

43. (y+\frac{1}{4})^2

44. (x+\frac{2}{3})^2

Answer

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

\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

25u^4+90u^2+81

49. (4y3−2)2

50. (8p3−3)2

Answer

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

64j^2−16

53. (11k+4)(11k−4)

54. (9c+5)(9c−5)

Answer

81c^2−25

55. (9c−2d)(9c+2d)

56. (7w+10x)(7w−10x)

Answer

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

p^2−\frac{16}{25}q^2

59. (ab−4)(ab+4)

60. (xy−9)(xy+9)

Answer

x^2y^2−81

61. (12p^3−11q^2)(12p^3+11q^2)

62. (15m^2−8n^4)(15m^2+8n^4)

Answer

225m^4−64n^8

In the following exercises, find each product.

63. (p−3)(p+3)

64. (t−9)^2

Answer

t^2−18t+81

65. (m+n)^2

66. (2x+y)(x−2y)

Answer

2x^2−3xy−2y^2

67. (2r+12)^2

68. (3p+8)(3p−8)

Answer

9p^2−64

69. (7a+b)(a−7b)

70. (k−6)^2

Answer

k^2−12k+36

71. (a^5−7b)^2

72. (x^2+8y)(8x−y^2)

Answer

8x^3−x^2y^2+64xy−8y^3

73. (r^6+s^6)(r^6−s^6)

74. (y^4+2z)^2

Answer

y^8+4y^4z+4z^2

75. (x^5+y^5)(x^5−y^5)

76. (m^3−8n)^2

Answer

m^6−16m^3n+64n^2

77. (9p+8q)^2

78. (r^2−s^3)(r^3+s^2)

Answer

r^5+r^2s^2−r^3s^3−s^5

Mixed Practice

79. (10y−6)+(4y−7)

80. (15p−4)+(3p−5)

Answer

18p−9

81. (x^2−4x−34)−(x^2+7x−6)

82. (j^2−8j−27)−(j^2+2j−12)

Answer

−10j−15

83. (\frac{1}{5}f^8)(20f^3)

84. (\frac{1}{4}d^5)(36d^2)

Answer

9d^7

85. (4a^3b)(9a^2b^6)

86. (6m^4n^3)(7mn^5)

Answer

72m^5n^8

87. −5m(m^2+3m−18)

88. 5q^3(q^2−2q+6)

Answer

5q^5−10q^4+30q^3

89. (s−7)(s+9)

90. (y^2−2y)(y+1)

Answer

y^3−y^2−2y

91. (5x−y)(x−4)

92. (6k−1)(k^2+2k−4)

Answer

6k^3−11k^2−26k+4

93. (3x−11y)(3x−11y)

94. (11−b)(11+b)

Answer

121−b^2

95. (rs−\frac{2}{7})(rs+\frac{2}{7})

96. (2x^2−3y^4)(2x^2+3y^4)

Answer

4x^4−9y^8

97. (m−15)^2

98. (3d+1)^2

Answer

9d^2+6d+1

99. (4a+10)^2

100. (3z+15)^2

Answer

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

(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

(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

(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

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?


This page titled 5.4E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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