Chapter Review Exercises
Determine the Degree of Polynomials
In the following exercises, determine the type of polynomial.
2. \(5m+9\)
- Answer
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binomial
4. \(y^2+6y^3+9y^4\)
- Answer
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other polynomial
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
6. \(−8y^3−5y^3\)
- Answer
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\(−13y^3\)
7. \((4a^2+9a−11)+(6a^2−5a+10)\)
8. \((8m^2+12m−5)−(2m^2−7m−1)\)
- Answer
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\(6m^2+19m−4\)
9. \((y^2−3y+12)+(5y^2−9)\)
10. \((5u^2+8u)−(4u−7)\)
- Answer
-
\(5u^2+4u+7\)
11. Find the sum of \(8q^3−27\) and \(q^2+6q−2\).
12. Find the difference of \(x^2+6x+8\) and \(x^2−8x+15\).
- Answer
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\(2x^2−2x+23\)
In the following exercises, simplify.
13. \(17mn^2−(−9mn^2)+3mn^2\)
14. \(18a−7b−21a\)
- Answer
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\(−7b−3a\)
16. \((6a^2+7)+(2a^2−5a−9)\)
- Answer
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\(8a^2−5a−2\)
17. \((3p^2−4p−9)+(5p^2+14)\)
18. \((7m^2−2m−5)−(4m^2+m−8)\)
- Answer
-
\(−3m+3\)
19. \((7b^2−4b+3)−(8b^2−5b−7)\)
20. Subtract \((8y^2−y+9)\) from \( (11y^2−9y−5) \)
- Answer
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\(3y^2−8y−14\)
21. Find the difference of \((z^2−4z−12)\) and \((3z^2+2z−11)\)
22. \((x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)\)
- Answer
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\(x^3+2x^2y−4xy^2\)
23. \((x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2)\)
Evaluate a Polynomial Function for a Given Value of the Variable
In the following exercises, find the function values for each polynomial function.
24. For the function \(f(x)=7x^2−3x+5\) find:
a. \(f(5)\) b. \(f(−2)\) c. \(f(0)\)
- Answer
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a. 165 b. 39 c. 5
25. For the function \(g(x)=15−16x^2\), find:
a. \(g(−1)\) b. \(g(0)\) c. \(g(2)\)
26. A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function \(h(t)=−16t^2+640\) gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when \(t=6\).
- Answer
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The height is 64 feet.
27. A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of \(p\) dollars each is given by the polynomial \(R(p)=−5p^2+360p\). Find the revenue received when \(p=110\) dollars.
Add and Subtract Polynomial Functions
In the following exercises, find a. \((f + g)(x)\) b. \((f + g)(3)\) c. \((f − g)(x\) d. \((f − g)(−2)\)
28. \(f(x)=2x^2−4x−7\) and \(g(x)=2x^2−x+5\)
- Answer
-
a. \((f+g)(x)=4x^2−5x−2\)
b. \((f+g)(3)=19\)
c. \((f−g)(x)=−3x−12\)
d. \((f−g)(−2)=−6\)
29. \(f(x)=4x^3−3x^2+x−1\) and \(g(x)=8x^3−1\)
Simplify Expressions Using the Properties for Exponents
In the following exercises, simplify each expression using the properties for exponents.
30. \(p^3·p^{10}\)
- Answer
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\(p^{13}\)
32. \(a·a^2·a^3\)
- Answer
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\(a^6\)
34. \(y^a·y^b\)
- Answer
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\(y^{a+b}\)
36. \(\dfrac{a^6}{a}\)
- Answer
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\(a^5\)
37. \(\dfrac{n^3}{n^{12}}\)
38. \(\dfrac{1}{x^5}\)
- Answer
-
\(\dfrac{1}{x^4}\)
42. \(12a^0−15b^0\)
- Answer
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\(−3\)
Use the Definition of a Negative Exponent
In the following exercises, simplify each expression.
44. \((−10)^{−3}\)
- Answer
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\(−\dfrac{1}{1000}\)
46. \((8n)^{−1}\)
- Answer
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\(\dfrac{1}{8n}\)
48. \(10^{−3}\)
- Answer
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\(\dfrac{1}{1000}\)
49. \(\dfrac{1}{a^{−4}}\)
50. \(\dfrac{1}{6^{−2}}\)
- Answer
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\(36\)
52. \( \left(−\dfrac{1}{5}\right)^{−3}\)
- Answer
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\(−\dfrac{1}{25}\)
54. \((−5)^{−3}\)
- Answer
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\(−\dfrac{1}{125}\)
55. \(\left(\dfrac{5}{9}\right)^{−2}\)
56. \(\left(−\dfrac{3}{x}\right)^{−3}\)
- Answer
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\(\dfrac{x^3}{27}\)
In the following exercises, simplify each expression using the Product Property.
58. \((3^2)^5\)
- Answer
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\(3^{10}\)
60. \(x^{−3}·x^9\)
- Answer
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\(x^5\)
62. \((uv^{−3})(u^{−4}v^{−2})\)
- Answer
-
\(\dfrac{1}{u^3v^5}\)
64. \(p^5·p^{−2}·p^{−4}\)
- Answer
-
\(\dfrac{1}{m^5}\)
In the following exercises, simplify each expression using the Power Property.
66. \(\dfrac{q^4}{q^{20}}\)
- Answer
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\(\dfrac{1}{q^{16}}\)
67. \(\dfrac{b^8}{b^{−2}}\)
68. \(\dfrac{n^{−3}}{n^{−5}}\)
- Answer
-
\(n^2\)
In the following exercises, simplify each expression using the Product to a Power Property.
70. \((−4pq)^0\)
- Answer
-
\(1\)
72. \((3y^{−4})^2\)
- Answer
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\(\dfrac{9}{y^8}\)
In the following exercises, simplify each expression using the Quotient to a Power Property.
73. \(\left(\dfrac{3}{5x}\right)^{−2}\)
74. \(\left(\dfrac{3xy^2}{z}\right)^4\)
- Answer
-
\(\dfrac{81x^4y^8}{z^4}\)
In the following exercises, simplify each expression by applying several properties.
76. \((x^2y)^2(3xy^5)^3\)
- Answer
-
\(27x^7y^{17}\)
77. \((−3a^{−2})^4(2a^4)^2(−6a^2)^3\)
78. \(\left(\dfrac{3xy^3}{4x^4y^{−2}}\right)^2\left(\dfrac{6xy^4}{8x^3y^{−2}}\right)^{−1}\)
- Answer
-
\(\dfrac{3y^4}{4x^4}\)
In the following exercises, write each number in scientific notation.
80. \(5,300,000\)
- Answer
-
\(5.3×10^6\)
In the following exercises, convert each number to decimal form.
82. \(2.9×10^4\)
- Answer
-
\(29,000\)
84. \(9.413×10^{−5}\)
- Answer
-
\(0.00009413\)
In the following exercises, multiply or divide as indicated. Write your answer in decimal form.
85. \((3×10^7)(2×10^{−4})\)
86. \((1.5×10^{−3})(4.8×10^{−1})\)
- Answer
-
\(0.00072\)
87. \(\dfrac{6×10^9}{2×10^{−1}}\)
88. \(\dfrac{9×10^{−3}}{1×10^{−6}}\)
- Answer
-
\(9,000\)
Multiply Monomials
In the following exercises, multiply the monomials.
90. \(\left(\frac{1}{3}c^2\right)(30c^8)\)
- Answer
-
\(10c^{10}\)
92. \( \left(\frac{2}{3}m^3n^6\right)\left(\frac{1}{6}m^4n^4\right)\)
- Answer
-
\(\dfrac{m^7n^{10}}{9}\)
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
94. \(a^2(a^2−9a−36)\)
- Answer
-
\(a^4−9a^3−36a^2\)
96. \((4n−5)(2n^3)\)
- Answer
-
\(8n^4−10n^3\)
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using:
a. the Distributive Property b. the FOIL method c. the Vertical Method.
98. \((y−4)(y+12)\)
- Answer
-
\(y^2+8y−48\)
100. \((6p−11)(3p−10)\)
- Answer
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\(18p^2−93p+110\)
In the following exercises, multiply the binomials. Use any method.
102. \((k+6)(k−9)\)
- Answer
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\(k^2−3k−54\)
104. \((2y−9)(5y−7)\)
- Answer
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\(10y^2−59y+63\)
106. \((x−8)(x+9)\)
- Answer
-
\(x^2+x−72\)
108. \((10a−1)(3a−3)\)
- Answer
-
\(30a^2−33a+3\)
Multiply a Polynomial by a Polynomial
In the following exercises, multiply using a. the Distributive Property b. the Vertical Method.
109. \((x+1)(x^2−3x−21)\)
110. \((5b−2)(3b^2+b−9)\)
- Answer
-
\(15b^3−b^2−47b+18\)
In the following exercises, multiply. Use either method.
111. \((m+6)(m^2−7m−30)\)
112. \((4y−1)(6y^2−12y+5)\)
- Answer
-
\(24y^2−54y^2+32y−5\)
Multiply Special Products
In the following exercises, square each binomial using the Binomial Squares Pattern.
114. \((x+\dfrac{3}{4})^2\)
- Answer
-
\(x^2+\dfrac{3}{2}x+\dfrac{9}{16}\)
116. \((5p+7q)^2\)
- Answer
-
\(25p^2+70pq+49q^2\)
In the following exercises, multiply each pair of conjugates using the Product of Conjugates.
118. \((6x+y)(6x−y)\)
- Answer
-
\(36x^2−y^2\)
119. \((a+\dfrac{2}3b)(a−\dfrac{2}{3}b)\)
120. \((12x^3−7y^2)(12x^3+7y^2)\)
- Answer
-
\(144x^6−49y^4\)
121. \((13a^2−8b4)(13a^2+8b^4)\)
Divide Monomials
In the following exercises, divide the monomials.
122. \(72p^{12}÷8p^3\)
- Answer
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\(9p^9\)
124. \(\dfrac{45y^6}{−15y^{10}}\)
- Answer
-
\(−3y^4\)
125. \(\dfrac{−30x^8}{−36x^9}\)
126. \(\dfrac{28a^9b}{7a^4b^3}\)
- Answer
-
\(\dfrac{4a^5}{b^2}\)
127. \(\dfrac{11u^6v^3}{55u^2v^8}\)
128. \(\dfrac{(5m^9n^3)(8m^3n^2)}{(10mn^4)(m^2n^5)}\)
- Answer
-
\(\dfrac{4m^9}{n^4}\)
129. \(\dfrac{(42r^2s^4)(54rs^2)}{(6rs^3)(9s)}\)
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial
130. \((54y^4−24y^3)÷(−6y^2)\)
- Answer
-
\(−9y^2+4y\)
131. \(\dfrac{63x^3y^2−99x^2y^3−45x^4y^3}{9x^2y^2}\)
132. \(\dfrac{12x^2+4x−3}{−4x}\)
- Answer
-
\(−3x−1+\dfrac{3}{4x}\)
Divide Polynomials using Long Division
In the following exercises, divide each polynomial by the binomial.
133. \((4x^2−21x−18)÷(x−6)\)
134. \((y^2+2y+18)÷(y+5)\)
- Answer
-
\(y−3+\dfrac{33}{q+6}\)
135. \((n^3−2n^2−6n+27)÷(n+3)\)
136. \((a^3−1)÷(a+1)\)
- Answer
-
\(a^2+a+1\)
Divide Polynomials using Synthetic Division
In the following exercises, use synthetic Division to find the quotient and remainder.
137. \(x^3−3x^2−4x+12\) is divided by \(x+2\)
138. \(2x^3−11x^2+11x+12\) is divided by \(x−3\)
- Answer
-
\(2x^2−5x−4;\space0\)
139. \(x^4+x^2+6x−10\) is divided by \(x+2\)
Divide Polynomial Functions
In the following exercises, divide.
140. For functions \(f(x)=x^2−15x+45\) and \(g(x)=x−9\), find a. \(\left(\dfrac{f}{g}\right)(x)\)
b. \(\left(\dfrac{f}{g}\right)(−2)\)
- Answer
-
a. \(\left(\dfrac{f}{g}\right)(x)=x−6\)
b. \(\left(\dfrac{f}{g}\right)(−2)=−8\)
141. For functions \(f(x)=x^3+x^2−7x+2\) and \(g(x)=x−2\), find a. \(\left(\dfrac{f}{g}\right)(x)\)
b. \(\left(\dfrac{f}{g}\right)(3)\)
Use the Remainder and Factor Theorem
In the following exercises, use the Remainder Theorem to find the remainder.
142. \(f(x)=x^3−4x−9\) is divided by \(x+2\)
- Answer
-
\(−9\)
143. \(f(x)=2x^3−6x−24\) divided by \(x−3\)
In the following exercises, use the Factor Theorem to determine if \(x−c\) is a factor of the polynomial function.
144. Determine whether \(x−2\) is a factor of \(x^3−7x^2+7x−6\)
- Answer
-
no
145. Determine whether \(x−3\) is a factor of \(x^3−7x^2+11x+3\)
Chapter Practice Test
1. For the polynomial \(8y^4−3y^2+1\)
a. Is it a monomial, binomial, or trinomial? b. What is its degree?
- Answer
-
a. trinomial b. 4
2. \((5a^2+2a−12)(9a^2+8a−4)\)
3. \((10x^2−3x+5)−(4x^2−6)\)
- Answer
-
\(6x^2−3x+11\)
4. \(\left(−\dfrac{3}{4}\right)^3\)
5. \(x^{−3}x^4\)
- Answer
-
\(x\)
7. \((47a^{18}b^{23}c^5)^0\)
- Answer
-
\(1\)
9. \((2y)^{−3}\)
- Answer
-
\(\dfrac{1}{8y^3}\)
11. \(\dfrac{x^4}{x^{−5}}\)
- Answer
-
\(x^9\)
13. \(\dfrac{24r^3s}{6r^2s^7}\)
- Answer
-
\(\dfrac{4r}{s^6}\)
15. \((8xy^3)(−6x^4y^6)\)
- Answer
-
\(−48x^5y^9\)
17. \((m+3)(7m−2)\)
- Answer
-
\(21m^2−19m−6\)
19. \((4x−3)^2\)
- Answer
-
\(16x^2−24x+9\)
21. \((15xy^3−35x^2y)÷5xy\)
- Answer
-
\(3y^2−7x \)
22. \((3x^3−10x^2+7x+10)÷(3x+2)\)
23. Use the Factor Theorem to determine if \(x+3\) a factor of \(x^3+8x^2+21x+18\).
- Answer
-
yes
24. a. Convert 112,000 to scientific notation.
b. Convert \(5.25×10^{−4}\) to decimal form.
In the following exercises, simplify and write your answer in decimal form.
25. \((2.4×10^8)(2×10^{−5})\)
- Answer
-
\(4.4×10^3\)
26. \(\dfrac{9×10^4}{3×10^{−1}}\)
27. For the function \(f(x)=6x^2−3x−9\) find:
a. \(f(3)\) b. \(f(−2)\) c. \(f(0)\)
- Answer
-
a. \(36\) b. \(21\) c. \(-9\)
28. For \(f(x)=2x^2−3x−5\) and \(g(x)=3x^2−4x+1\), find
a. \((f+g)(x)\) b. \((f+g)(1)\)
c. \((f−g)(x)\) d. \((f−g)(−2)\)
29. For functions \(f(x)=3x^2−23x−36\) and \(g(x)=x−9\), find
a. \(\left(\dfrac{f}{g}\right)(x)\) b. \(\left(\dfrac{f}{g}\right)(3)\)
- Answer
-
a. \(\left(\dfrac{f}{g}\right)(x)=3x+4\)
b. \(\left(\dfrac{f}{g}\right)(3)=13\)
30. A hiker drops a pebble from a bridge \(240\) feet above a canyon. The function \(h(t)=−16t^2+240\) gives the height of the pebble \(t\) seconds after it was dropped. Find the height when \(t=3\).
