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.
10. \((y+9)(y+3)\)
- Answer
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\(y^2+12y+27\)
12. \((7q+4)(3q−8)\)
- Answer
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\(21q^2−44q−32\)
In the following exercises, multiply the binomials. Use any method.
14. \((y−6)(y−2)\)
- Answer
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\(y^2−8y+12\)
16. \((6p+5)(p+1)\)
- Answer
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\(6p^2+11p+5\)
18. \((m+11)(m−4)\)
- Answer
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\(m^2+7m−44\)
20. \((3r−8)(11r+1)\)
- Answer
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\(33r^2−85r−8\)
22. \((y^2−4)(y+3)\)
- Answer
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\(y^3+3y^2−4y−12\)
24. \((2xy+3)(3xy+2)\)
- Answer
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\(6x^2y^2+13xy+6\)
26. \((y^2−7)(y^2−4)\)
- Answer
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\(y^4−11y^2+28\)
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.
30. \((u+4)(u^2+3u+2)\)
- Answer
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\(u^3+7u^2+14u+8\)
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.
36. \((p−4)(p^2−6p+9)\)
- Answer
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\(p^3−10p^2+33p−36\)
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.
40. \((q+12)^2\)
- Answer
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\(q^2+24q+144\)
42. \((2y−3z)^2\)
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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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\(\frac{1}{64}x^2−\frac{1}{36}xy+\frac{1}{81}y^2\)
48. \((5u^2+9)^2\)
- Answer
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\(25u^4+90u^2+81\)
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.
52. \((8j+4)(8j−4)\)
- Answer
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\(64j^2−16\)
54. \((9c+5)(9c−5)\)
- Answer
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\(81c^2−25\)
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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\(p^2−\frac{16}{25}q^2\)
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.
64. \((t−9)^2\)
- Answer
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\(t^2−18t+81\)
66. \((2x+y)(x−2y)\)
- Answer
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\(2x^2−3xy−2y^2\)
68. \((3p+8)(3p−8)\)
- Answer
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\(9p^2−64\)
70. \((k−6)^2\)
- Answer
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\(k^2−12k+36\)
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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\(m^6−16m^3n+64n^2\)
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
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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\(9d^7\)
86. \((6m^4n^3)(7mn^5)\)
- Answer
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\(72m^5n^8\)
88. \(5q^3(q^2−2q+6)\)
- Answer
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\(5q^5−10q^4+30q^3\)
90. \((y^2−2y)(y+1)\)
- Answer
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\(y^3−y^2−2y\)
92. \((6k−1)(k^2+2k−4)\)
- Answer
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\(6k^3−11k^2−26k+4\)
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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\(4x^4−9y^8\)
98. \((3d+1)^2\)
- Answer
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\(9d^2+6d+1\)
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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ⓐ \((f·g)(x)=x^4+6x^3+9x^2+10x−12\) ⓑ \((f·g)(1)=14\)