5.3E: Exercises

Last updated
Save as PDF

Page ID: 30854

Lynn Marecek
Professor (Mathematics) at Santa Ana College
Publisher: OpenStax CNX

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

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.

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