# Exercises: Multiply Polynomials (5.3)

## Exercises: Multiply a Polynomial by a Monomial

Instructions: For questions 1-64, multiply.
1. $4\left(w+10\right)$
Solution

$4w+40$

2. $6\left(b+8\right)$

3. $-3\left(a+7\right)$
Solution

$-3a-21$

4. $-5\left(p+9\right)$

5. $2\left(x-7\right)$
Solution

$2x-14$

6. $7\left(y-4\right)$

7. $-3\left(k-4\right)$
Solution

$-3k+12$

8. $-8\left(j-5\right)$

9. $q\left(q+5\right)$
Solution

${q}^{2}+5q$

10. $k\left(k+7\right)$

11. $-b\left(b+9\right)$
Solution

$-{b}^{2}-9b$

12. $-y\left(y+3\right)$

13. $-x\left(x-10\right)$
Solution

$-{x}^{2}+10x$

14. $-p\left(p-15\right)$

15. $6r\left(4r+s\right)$
Solution

$24{r}^{2}+6rs$

16. $5c\left(9c+d\right)$

17. $12x\left(x-10\right)$
Solution

$12{x}^{2}-120x$

18. $9m\left(m-11\right)$

19. $-9a\left(3a+5\right)$
Solution

$-27{a}^{2}-45a$

20. $-4p\left(2p+7\right)$

21. $3\left({p}^{2}+10p+25\right)$
Solution

$3{p}^{2}+30p+75$

22. $6\left({y}^{2}+8y+16\right)$

23. $-8x\left({x}^{2}+2x-15\right)$
Solution

$-8{x}^{3}-16{x}^{2}+120x$

24. $-5t\left({t}^{2}+3t-18\right)$

25. $5{q}^{3}\left({q}^{3}-2q+6\right)$
Solution

$5{q}^{6}-10{q}^{4}+30{q}^{3}$

26. $4{x}^{3}\left({x}^{4}-3x+7\right)$

27. $-8y\left({y}^{2}+2y-15\right)$
Solution

$-8{y}^{3}-16{y}^{2}+120y$

28. $-5m\left({m}^{2}+3m-18\right)$

29. $5{q}^{3}\left({q}^{2}-2q+6\right)$
Solution

$5{q}^{5}-10{q}^{4}+30{q}^{3}$

30. $9{r}^{3}\left({r}^{2}-3r+5\right)$

31. $-4{z}^{2}\left(3{z}^{2}+12z-1\right)$
Solution

$-12{z}^{4}-48{z}^{3}+4{z}^{2}$

32. $-3{x}^{2}\left(7{x}^{2}+10x-1\right)$

33. $\left(2m-9\right)m$
Solution

$2{m}^{2}-9m$

34. $\left(8j-1\right)j$

35. $\left(w-6\right)\cdot 8$
Solution

$8w-48$

36. $\left(k-4\right)\cdot 5$

37. $4\left(x+10\right)$
Solution

$4x+40$

38. $6\left(y+8\right)$

39. $15\left(r-24\right)$
Solution

$15r-360$

40. $12\left(v-30\right)$

41. $-3\left(m+11\right)$
Solution

$-3m-33$

42. $-4\left(p+15\right)$

43. $-8\left(z-5\right)$
Solution

$-8z+40$

44. $-3\left(x-9\right)$

45. $u\left(u+5\right)$
Solution

${u}^{2}+5u$

46. $q\left(q+7\right)$

47. $n\left({n}^{2}-3n\right)$
Solution

${n}^{3}-3{n}^{2}$

48. $s\left({s}^{2}-6s\right)$

49. $6x\left(4x+y\right)$
Solution

$24{x}^{2}+6xy$

50. $5a\left(9a+b\right)$

51. $5p\left(11p-5q\right)$
Solution

$55{p}^{2}-25pq$

52. $12u\left(3u-4v\right)$

53. $3\left({v}^{2}+10v+25\right)$
Solution

$3{v}^{2}+30v+75$

54. $6\left({x}^{2}+8x+16\right)$

55. $2n\left(4{n}^{2}-4n+1\right)$
Solution

$8{n}^{3}-8{n}^{2}+2n$

56. $3r\left(2{r}^{2}-6r+2\right)$

57. $-8y\left({y}^{2}+2y-15\right)$
Solution

$-8{y}^{3}-16{y}^{2}+120y$

58. $-5m\left({m}^{2}+3m-18\right)$

59. $5{q}^{3}\left({q}^{2}-2q+6\right)$
Solution

$5{q}^{5}-10{q}^{4}+30{q}^{3}$

60. $9{r}^{3}\left({r}^{2}-3r+5\right)$

61. $-4{z}^{2}\left(3{z}^{2}+12z-1\right)$
Solution

$-12{z}^{4}-48{z}^{3}+4{z}^{2}$

62. $-3{x}^{2}\left(7{x}^{2}+10x-1\right)$

63. $\left(2y-9\right)y$
Solution

$18{y}^{2}-9y$

64. $\left(8b-1\right)b$

## Exercises: Multiply a Binomial by a Binomial

Instructions: For questions 65-68, multiply the following binomials using:

a. the Distributive Property
b. the FOIL method
c. the Vertical Method.

65. $\left(w+5\right)\left(w+7\right)$
Solution

${w}^{2}+12w+35$

66. $\left(y+9\right)\left(y+3\right)$

67. $\left(p+11\right)\left(p-4\right)$
Solution

${p}^{2}+7p-44$

68. $\left(q+4\right)\left(q-8\right)$

## Exercises: Multiply a Binomial by a Binomial

Instructions: For questions 69-94, multiply the binomials. Use any method.
69. $\left(x+8\right)\left(x+3\right)$
Solution

${x}^{2}+11x+24$

70. $\left(y+7\right)\left(y+4\right)$

71. $\left(y-6\right)\left(y-2\right)$
Solution

${y}^{2}-8y+12$

72. $\left(x-7\right)\left(x-2\right)$

73. $\left(w-4\right)\left(w+7\right)$
Solution

${w}^{2}+3w-28$

74. $\left(q-5\right)\left(q+8\right)$

75. $\left(p+12\right)\left(p-5\right)$
Solution

${p}^{2}+7p-60$

76. $\left(m+11\right)\left(m-4\right)$

77. $\left(6p+5\right)\left(p+1\right)$
Solution

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

78. $\left(7m+1\right)\left(m+3\right)$

79. $\left(2t-9\right)\left(10t+1\right)$
Solution

$20{t}^{2}-88t-9$

80. $\left(3r-8\right)\left(11r+1\right)$

81. $\left(5x-y\right)\left(3x-6\right)$
Solution

$15{x}^{2}-3xy-30x+6y$

82. $\left(10a-b\right)\left(3a-4\right)$

83. $\left(a+b\right)\left(2a+3b\right)$
Solution

$2{a}^{2}+5ab+3{b}^{2}$

84. $\left(r+s\right)\left(3r+2s\right)$

85. $\left(4z-y\right)\left(z-6\right)$
Solution

$4{z}^{2}-24z-zy+6y$

86. $\left(5x-y\right)\left(x-4\right)$

87. $\left({x}^{2}+3\right)\left(x+2\right)$
Solution

${x}^{3}+2{x}^{2}+3x+6$

88. $\left({y}^{2}-4\right)\left(y+3\right)$

89. $\left({x}^{2}+8\right)\left({x}^{2}-5\right)$
Solution

${x}^{4}+3{x}^{2}-40$

90. $\left({y}^{2}-7\right)\left({y}^{2}-4\right)$

91. $\left(5ab-1\right)\left(2ab+3\right)$
Solution

$10{a}^{2}{b}^{2}+13ab-3$

92. $\left(2xy+3\right)\left(3xy+2\right)$

93. $\left(6pq-3\right)\left(4pq-5\right)$
Solution

$24{p}^{2}{q}^{2}-42pq+15$

94. $\left(3rs-7\right)\left(3rs-4\right)$

## Exercises: Multiply a Trinomial by a Binomial

Instructions: For questions 95-98, multiply using:

a. the Distributive Property
b. the Vertical Method

95. $\left(x+5\right)\left({x}^{2}+4x+3\right)$
Solution

${x}^{3}+9{x}^{2}+23x+15$

96. $\left(u+4\right)\left({u}^{2}+3u+2\right)$

97. $\left(y+8\right)\left(4{y}^{2}+y-7\right)$
Solution

$4{y}^{3}+33{y}^{2}+y-56$

98. $\left(a+10\right)\left(3{a}^{2}+a-5\right)$

## Exercises: Multiply a Trinomial by a Binomial

Instructions: For questions 99-102, multiply. Use either method.
99. $\left(w-7\right)\left({w}^{2}-9w+10\right)$
Solution

${w}^{3}-16{w}^{2}+73w-70$

100. $\left(p-4\right)\left({p}^{2}-6p+9\right)$

101. $\left(3q+1\right)\left({q}^{2}-4q-5\right)$
Solution

$3{q}^{3}-11{q}^{2}-19q-5$

102. $\left(6r+1\right)\left({r}^{2}-7r-9\right)$

## Exercises: Mixed Practice

Instructions: For questions 103-121, solve.

103. $\left(10y-6\right)+\left(4y-7\right)$
Solution

$14y-13$

104. $\left(15p-4\right)+\left(3p-5\right)$

105. $\left({x}^{2}-4x-34\right)-\left({x}^{2}+7x-6\right)$
Solution

$-11x-28$

106. $\left({j}^{2}-8j-27\right)-\left({j}^{2}+2j-12\right)$

107. $5q\left(3{q}^{2}-6q+11\right)$
Solution

$15{q}^{3}-30{q}^{2}+55q$

108. $8t\left(2{t}^{2}-5t+6\right)$

109. $\left(s-7\right)\left(s+9\right)$
Solution

${s}^{2}+2s-63$

110. $\left(x-5\right)\left(x+13\right)$

111. $\left({y}^{2}-2y\right)\left(y+1\right)$
Solution

${y}^{3}-{y}^{2}-2y$

112. $\left({a}^{2}-3a\right)\left(4a+5\right)$

113. $\left(3n-4\right)\left({n}^{2}+n-7\right)$
Solution

$3{n}^{3}-{n}^{2}-25n+28$

114. $\left(6k-1\right)\left({k}^{2}+2k-4\right)$

115. $\left(7p+10\right)\left(7p-10\right)$
Solution

$49{p}^{2}-100$

116. $\left(3y+8\right)\left(3y-8\right)$

117. $\left(4{m}^{2}-3m-7\right){m}^{2}$
Solution

$4{m}^{4}-3{m}^{3}-7{m}^{2}$

118. $\left(15{c}^{2}-4c+5\right){c}^{4}$

119. $\left(5a+7b\right)\left(5a+7b\right)$
Solution

$25{a}^{2}+70ab+49{b}^{2}$

120. $\left(3x-11y\right)\left(3x-11y\right)$

121. $\left(4y+12z\right)\left(4y-12z\right)$
Solution

$16{y}^{2}-144{z}^{2}$

## Exercises: Everyday Math

Instructions: For questions 122-123, solve the given everyday math word problems.

122. Mental math. You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply $13$ times $15$. Think of $13$ as $10+3$ and $15$ as $10+5$.

a. Multiply $\left(10+3\right)\left(10+5\right)$ by the FOIL method.
b. Multiply $13\cdot 15$ without using a calculator.
c. Which way is easier for you? Why?

123. Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply $18$ times $17$. Think of $18$ as $20-2$ and $17$ as $20-3$.

a. Multiply $\left(20-2\right)\left(20-3\right)$ by the FOIL method.
b. Multiply $18\cdot 17$ without using a calculator.
c. Which way is easier for you? Why?

Solution

a. $306$
b. $306$
c. Answers will vary.

## Exercises: Writing Exercises

Instructions: For questions 124-129, answer the given writing exercises.
124. Which method do you prefer to use when multiplying two binomials: the Distributive Property, the FOIL method, or the Vertical Method? Why?

125. Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?
Solution

126. Multiply the following:$\begin{array}{c}\left(x+2\right)\left(x-2\right)\\ \left(y+7\right)\left(y-7\right)\\ \left(w+5\right)\left(w-5\right)\end{array}$

Explain the pattern that you see in your answers.

127. Multiply the following:

$\begin{array}{c}\left(m-3\right)\left(m+3\right)\\ \left(n-10\right)\left(n+10\right)\\ \left(p-8\right)\left(p+8\right)\end{array}$

Explain the pattern that you see in your answers.

Solution

128. Multiply the following:

$\begin{array}{c}\left(p+3\right)\left(p+3\right)\\ \left(q+6\right)\left(q+6\right)\\ \left(r+1\right)\left(r+1\right)\end{array}$

Explain the pattern that you see in your answers.

129. Multiply the following:

$\begin{array}{c}\left(x-4\right)\left(x-4\right)\\ \left(y-1\right)\left(y-1\right)\\ \left(z-7\right)\left(z-7\right)\end{array}$

Explain the pattern that you see in your answers.

Solution