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# 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. ⓐ $$(−10x^5)(−3x^3)$$ ⓑ $$(58x^3y)(24x^5y)$$

ⓐ$$30x^8$$ ⓑ $$15x^8y^2$$

3. ⓐ $$(−8u^6)(−9u)$$ ⓑ $$(\frac{2}{3}x^2y)(\frac{3}{4}xy^2)$$

4. ⓐ $$(−6c^4)(−12c)$$ ⓑ $$(\frac{3}{5}m^3n^2)(\frac{5}{9}m^2n^3)$$

ⓐ $$72c^5$$ ⓑ $$\frac{1}{3}m^5n^5$$

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. ⓐ $$−5t(t^2+3t−18)$$ ⓑ $$9r^3s(r^2−3rs+5s^2)$$

ⓐ $$−5t^3−15t^2+90t$$
ⓑ $$9sr^5−27s^2r^4+45s^3r^3$$

7. ⓐ $$−8y(y^2+2y−15)$$ ⓑ $$−4y^2z^2(3y^2+12yz−z^2)$$

8. ⓐ $$−5m(m^2+3m−18)$$ ⓑ $$−3x^2y^2(7x^2+10xy−y^2)$$

ⓐ $$−5m^3−15m^2+90m$$
ⓑ $$−21x^4y^2−30x^3y^3+3x^2y^4$$

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. $$(y+9)(y+3)$$

$$y^2+12y+27$$

11. $$(4p+11)(5p−4)$$

12. $$(7q+4)(3q−8)$$

$$21q^2−44q−32$$

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

13. $$(x+8)(x+3)$$

14. $$(y−6)(y−2)$$

$$y^2−8y+12$$

15. $$(2t−9)(10t+1)$$

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

$$6p^2+11p+5$$

17. $$(q−5)(q+8)$$

18. $$(m+11)(m−4)$$

$$m^2+7m−44$$

19. $$(7m+1)(m−3)$$

20. $$(3r−8)(11r+1)$$

$$33r^2−85r−8$$

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

22. $$(y^2−4)(y+3)$$

$$y^3+3y^2−4y−12$$

23. $$(5ab−1)(2ab+3)$$

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

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

25. $$(x^2+8)(x^2−5)$$

26. $$(y^2−7)(y^2−4)$$

$$y^4−11y^2+28$$

27. $$(6pq−3)(4pq−5)$$

28. $$(3rs−7)(3rs−4)$$

$$9r^2s^2−33rs+28$$

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. $$(u+4)(u^2+3u+2)$$

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

31. $$(y+8)(4y^2+y−7)$$

32. $$(a+10)(3a^2+a−5)$$

$$3a^3+31a^2+5a−50$$

33. $$(y^2−3y+8)(4y^2+y−7)$$

34. $$(2a^2−5a+10)(3a^2+a−5)$$

$$6a^4−13a^3+15a^2+35a−50$$

Multiply Special Products

In the following exercises, multiply. Use either method.

35. $$(w−7)(w^2−9w+10)$$

36. $$(p−4)(p^2−6p+9)$$

$$p^3−10p^2+33p−36$$

37. $$(3q+1)(q^2−4q−5)$$

38. $$(6r+1)(r^2−7r−9)$$

$$6r^3−41r^2−61r−9$$

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

39. $$(w+4)^2$$

40. $$(q+12)^2$$

$$q^2+24q+144$$

41. $$(3x−y)^2$$

42. $$(2y−3z)^2$$

$$4y^2−12yz+9z^2$$

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

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

$$x^2+\frac{4}{3}x+\frac{4}{9}$$

45. $$(\frac{1}{5}x−\frac{1}{7}y)^2$$

46. $$(\frac{1}{8}x−\frac{1}{9}y)^2$$

$$\frac{1}{64}x^2−\frac{1}{36}xy+\frac{1}{81}y^2$$

47. $$(3x^2+2)^2$$

48. $$(5u^2+9)^2$$

$$25u^4+90u^2+81$$

49. $$(4y3−2)2$$

50. $$(8p3−3)2$$

$$64p^6−48p^3+9$$

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

51. $$(5k+6)(5k−6)$$

52. $$(8j+4)(8j−4)$$

$$64j^2−16$$

53. $$(11k+4)(11k−4)$$

54. $$(9c+5)(9c−5)$$

$$81c^2−25$$

55. $$(9c−2d)(9c+2d)$$

56. $$(7w+10x)(7w−10x)$$

$$49w^2−100x^2$$

57. $$(m+\frac{2}{3}n)(m−\frac{2}{3}n)$$

58. $$(p+\frac{4}{5}q)(p−\frac{4}{5}q)$$

$$p^2−\frac{16}{25}q^2$$

59. $$(ab−4)(ab+4)$$

60. $$(xy−9)(xy+9)$$

$$x^2y^2−81$$

61. $$(12p^3−11q^2)(12p^3+11q^2)$$

62. $$(15m^2−8n^4)(15m^2+8n^4)$$

$$225m^4−64n^8$$

In the following exercises, find each product.

63. $$(p−3)(p+3)$$

64. $$(t−9)^2$$

$$t^2−18t+81$$

65. $$(m+n)^2$$

66. $$(2x+y)(x−2y)$$

$$2x^2−3xy−2y^2$$

67. $$(2r+12)^2$$

68. $$(3p+8)(3p−8)$$

$$9p^2−64$$

69. $$(7a+b)(a−7b)$$

70. $$(k−6)^2$$

$$k^2−12k+36$$

71. $$(a^5−7b)^2$$

72. $$(x^2+8y)(8x−y^2)$$

$$8x^3−x^2y^2+64xy−8y^3$$

73. $$(r^6+s^6)(r^6−s^6)$$

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

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

75. $$(x^5+y^5)(x^5−y^5)$$

76. $$(m^3−8n)^2$$

$$m^6−16m^3n+64n^2$$

77. $$(9p+8q)^2$$

78. $$(r^2−s^3)(r^3+s^2)$$

$$r^5+r^2s^2−r^3s^3−s^5$$

Mixed Practice

79. $$(10y−6)+(4y−7)$$

80. $$(15p−4)+(3p−5)$$

$$18p−9$$

81. $$(x^2−4x−34)−(x^2+7x−6)$$

82. $$(j^2−8j−27)−(j^2+2j−12)$$

$$−10j−15$$

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

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

$$9d^7$$

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

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

$$72m^5n^8$$

87. $$−5m(m^2+3m−18)$$

88. $$5q^3(q^2−2q+6)$$

$$5q^5−10q^4+30q^3$$

89. $$(s−7)(s+9)$$

90. $$(y^2−2y)(y+1)$$

$$y^3−y^2−2y$$

91. $$(5x−y)(x−4)$$

92. $$(6k−1)(k^2+2k−4)$$

$$6k^3−11k^2−26k+4$$

93. $$(3x−11y)(3x−11y)$$

94. $$(11−b)(11+b)$$

$$121−b^2$$

95. $$(rs−\frac{2}{7})(rs+\frac{2}{7})$$

96. $$(2x^2−3y^4)(2x^2+3y^4)$$

$$4x^4−9y^8$$

97. $$(m−15)^2$$

98. $$(3d+1)^2$$

$$9d^2+6d+1$$

99. $$(4a+10)^2$$

100. $$(3z+15)^2$$

$$9z^2−\frac{6}{5}z+\frac{1}{25}$$

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 ⓐ $$(f·g)(x)$$ ⓑ $$(f·g)(−2)$$

ⓐ $$(f·g)(x)=4x^3−x^2−8x+5$$
ⓑ $$(f·g)(−2)=−15$$

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 ⓐ $$(f·g)(x)$$ ⓑ $$(f·g)(−2)$$

ⓐ $$(f·g)(x)=49x^2−64$$
ⓑ $$(f·g)(−2)=187$$

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 ⓐ $$(f·g)(x)$$ ⓑ $$(f·g)(1)$$

ⓐ $$(f·g)(x)=x^4+6x^3+9x^2+10x−12$$ ⓑ $$(f·g)(1)=14$$

## 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:

$$(x+2)(x−2)$$

$$(y+7)(y−7)$$

$$(w+5)(w−5)$$

109. Multiply the following:

$$(p+3)(p+3)$$

$$(q+6)(q+6)$$

$$(r+1)(r+1)$$

110. Why does $$(a+b)^2$$ result in a trinomial, but $$(a−b)(a+b)$$ result in a binomial?