5.3E: Exercises
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
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ⓐ\(30x^8\) ⓑ \(15x^8y^2\)
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
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ⓐ \(72c^5\) ⓑ \(\frac{1}{3}m^5n^5\)
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
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ⓐ \(−5t^3−15t^2+90t\)
ⓑ \(9sr^5−27s^2r^4+45s^3r^3\)
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
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ⓐ \(−5m^3−15m^2+90m\)
ⓑ \(−21x^4y^2−30x^3y^3+3x^2y^4\)
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
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\(y^2+12y+27\)
11. \((4p+11)(5p−4)\)
12. \((7q+4)(3q−8)\)
- Answer
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\(21q^2−44q−32\)
In the following exercises, multiply the binomials. Use any method.
13. \((x+8)(x+3)\)
14. \((y−6)(y−2)\)
- Answer
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\(y^2−8y+12\)
15. \((2t−9)(10t+1)\)
16. \((6p+5)(p+1)\)
- Answer
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\(6p^2+11p+5\)
17. \((q−5)(q+8)\)
18. \((m+11)(m−4)\)
- Answer
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\(m^2+7m−44\)
19. \((7m+1)(m−3)\)
20. \((3r−8)(11r+1)\)
- Answer
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\(33r^2−85r−8\)
21. \((x^2+3)(x+2)\)
22. \((y^2−4)(y+3)\)
- Answer
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\(y^3+3y^2−4y−12\)
23. \((5ab−1)(2ab+3)\)
24. \((2xy+3)(3xy+2)\)
- Answer
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\(6x^2y^2+13xy+6\)
25. \((x^2+8)(x^2−5)\)
26. \((y^2−7)(y^2−4)\)
- Answer
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\(y^4−11y^2+28\)
27. \((6pq−3)(4pq−5)\)
28. \((3rs−7)(3rs−4)\)
- Answer
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\(9r^2s^2−33rs+28\)
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
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\(u^3+7u^2+14u+8\)
31. \((y+8)(4y^2+y−7)\)
32. \((a+10)(3a^2+a−5)\)
- Answer
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\(3a^3+31a^2+5a−50\)
33. \((y^2−3y+8)(4y^2+y−7)\)
34. \((2a^2−5a+10)(3a^2+a−5)\)
- Answer
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\(6a^4−13a^3+15a^2+35a−50\)
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
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\(p^3−10p^2+33p−36\)
37. \((3q+1)(q^2−4q−5)\)
38. \((6r+1)(r^2−7r−9)\)
- Answer
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\(6r^3−41r^2−61r−9\)
In the following exercises, square each binomial using the Binomial Squares Pattern.
39. \((w+4)^2\)
40. \((q+12)^2\)
- Answer
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\(q^2+24q+144\)
41. \((3x−y)^2\)
42. \((2y−3z)^2\)
- Answer
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\(4y^2−12yz+9z^2\)
43. \((y+\frac{1}{4})^2\)
44. \((x+\frac{2}{3})^2\)
- Answer
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\(x^2+\frac{4}{3}x+\frac{4}{9}\)
45. \((\frac{1}{5}x−\frac{1}{7}y)^2\)
46. \((\frac{1}{8}x−\frac{1}{9}y)^2\)
- Answer
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\(\frac{1}{64}x^2−\frac{1}{36}xy+\frac{1}{81}y^2\)
47. \((3x^2+2)^2\)
48. \((5u^2+9)^2\)
- Answer
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\(25u^4+90u^2+81\)
49. \((4y3−2)2\)
50. \((8p3−3)2\)
- Answer
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\(64p^6−48p^3+9\)
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
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\(64j^2−16\)
53. \((11k+4)(11k−4)\)
54. \((9c+5)(9c−5)\)
- Answer
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\(81c^2−25\)
55. \((9c−2d)(9c+2d)\)
56. \((7w+10x)(7w−10x)\)
- Answer
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\(49w^2−100x^2\)
57. \((m+\frac{2}{3}n)(m−\frac{2}{3}n)\)
58. \((p+\frac{4}{5}q)(p−\frac{4}{5}q)\)
- Answer
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\(p^2−\frac{16}{25}q^2\)
59. \((ab−4)(ab+4)\)
60. \((xy−9)(xy+9)\)
- Answer
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\(x^2y^2−81\)
61. \((12p^3−11q^2)(12p^3+11q^2)\)
62. \((15m^2−8n^4)(15m^2+8n^4)\)
- Answer
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\(225m^4−64n^8\)
In the following exercises, find each product.
63. \((p−3)(p+3)\)
64. \((t−9)^2\)
- Answer
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\(t^2−18t+81\)
65. \((m+n)^2\)
66. \((2x+y)(x−2y)\)
- Answer
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\(2x^2−3xy−2y^2\)
67. \((2r+12)^2\)
68. \((3p+8)(3p−8)\)
- Answer
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\(9p^2−64\)
69. \((7a+b)(a−7b)\)
70. \((k−6)^2\)
- Answer
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\(k^2−12k+36\)
71. \((a^5−7b)^2\)
72. \((x^2+8y)(8x−y^2)\)
- Answer
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\(8x^3−x^2y^2+64xy−8y^3\)
73. \((r^6+s^6)(r^6−s^6)\)
74. \((y^4+2z)^2\)
- Answer
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\(y^8+4y^4z+4z^2\)
75. \((x^5+y^5)(x^5−y^5)\)
76. \((m^3−8n)^2\)
- Answer
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\(m^6−16m^3n+64n^2\)
77. \((9p+8q)^2\)
78. \((r^2−s^3)(r^3+s^2)\)
- Answer
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\(r^5+r^2s^2−r^3s^3−s^5\)
Mixed Practice
79. \((10y−6)+(4y−7)\)
80. \((15p−4)+(3p−5)\)
- Answer
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\(18p−9\)
81. \((x^2−4x−34)−(x^2+7x−6)\)
82. \((j^2−8j−27)−(j^2+2j−12)\)
- Answer
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\(−10j−15\)
83. \((\frac{1}{5}f^8)(20f^3)\)
84. \((\frac{1}{4}d^5)(36d^2)\)
- Answer
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\(9d^7\)
85. \((4a^3b)(9a^2b^6)\)
86. \((6m^4n^3)(7mn^5)\)
- Answer
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\(72m^5n^8\)
87. \(−5m(m^2+3m−18)\)
88. \(5q^3(q^2−2q+6)\)
- Answer
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\(5q^5−10q^4+30q^3\)
89. \((s−7)(s+9)\)
90. \((y^2−2y)(y+1)\)
- Answer
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\(y^3−y^2−2y\)
91. \((5x−y)(x−4)\)
92. \((6k−1)(k^2+2k−4)\)
- Answer
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\(6k^3−11k^2−26k+4\)
93. \((3x−11y)(3x−11y)\)
94. \((11−b)(11+b)\)
- Answer
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\(121−b^2\)
95. \((rs−\frac{2}{7})(rs+\frac{2}{7})\)
96. \((2x^2−3y^4)(2x^2+3y^4)\)
- Answer
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\(4x^4−9y^8\)
97. \((m−15)^2\)
98. \((3d+1)^2\)
- Answer
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\(9d^2+6d+1\)
99. \((4a+10)^2\)
100. \((3z+15)^2\)
- Answer
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\(9z^2−\frac{6}{5}z+\frac{1}{25}\)
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
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ⓐ \((f·g)(x)=4x^3−x^2−8x+5\)
ⓑ \((f·g)(−2)=−15\)
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
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ⓐ \((f·g)(x)=49x^2−64\)
ⓑ \((f·g)(−2)=187\)
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
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ⓐ \((f·g)(x)=x^4+6x^3+9x^2+10x−12\) ⓑ \((f·g)(1)=14\)
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
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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
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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?