5.4E: Exercises

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Feb 14, 2022
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- 5.4: Multiply Polynomials
- 5.5: Dividing Polynomials

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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: ⓐ $30x^8$ ⓑ $15x^8y^2$

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: ⓐ $72c^5$ ⓑ $\frac{1}{3}m^5n^5$

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.

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ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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