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# Chapter 5 Review Exercises

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• OpenStax
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## Chapter Review Exercises

Determine the Degree of Polynomials

In the following exercises, determine the type of polynomial.

1. $$16x^2−40x−25$$

2. $$5m+9$$

binomial

3. $$−15$$

4. $$y^2+6y^3+9y^4$$

other polynomial

In the following exercises, add or subtract the polynomials.

5. $$4p+11p$$

6. $$−8y^3−5y^3$$

$$−13y^3$$

7. $$(4a^2+9a−11)+(6a^2−5a+10)$$

8. $$(8m^2+12m−5)−(2m^2−7m−1)$$

$$6m^2+19m−4$$

9. $$(y^2−3y+12)+(5y^2−9)$$

10. $$(5u^2+8u)−(4u−7)$$

$$5u^2+4u+7$$

11. Find the sum of $$8q^3−27$$ and $$q^2+6q−2$$.

12. Find the difference of $$x^2+6x+8$$ and $$x^2−8x+15$$.

$$2x^2−2x+23$$

In the following exercises, simplify.

13. $$17mn^2−(−9mn^2)+3mn^2$$

14. $$18a−7b−21a$$

$$−7b−3a$$

15. $$2pq^2−5p−3q^2$$

16. $$(6a^2+7)+(2a^2−5a−9)$$

$$8a^2−5a−2$$

17. $$(3p^2−4p−9)+(5p^2+14)$$

18. $$(7m^2−2m−5)−(4m^2+m−8)$$

$$−3m+3$$

19. $$(7b^2−4b+3)−(8b^2−5b−7)$$

20. Subtract $$(8y^2−y+9)$$ from $$(11y^2−9y−5)$$

$$3y^2−8y−14$$

21. Find the difference of $$(z^2−4z−12)$$ and $$(3z^2+2z−11)$$

22. $$(x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)$$

$$x^3+2x^2y−4xy^2$$

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. $$f(0)$$

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 $$t=6$$.

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.

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 $$g(x)=2x^2−x+5$$

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 $$g(x)=8x^3−1$$

### Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

30. $$p^3·p^{10}$$

$$p^{13}$$

31. $$2·2^6$$

32. $$a·a^2·a^3$$

$$a^6$$

33. $$x·x^8$$

34. $$y^a·y^b$$

$$y^{a+b}$$

35. $$\dfrac{2^8}{2^2}$$

36. $$\dfrac{a^6}{a}$$

$$a^5$$

37. $$\dfrac{n^3}{n^{12}}$$

38. $$\dfrac{1}{x^5}$$

$$\dfrac{1}{x^4}$$

39. $$3^0$$

40. $$y^0$$

$$1$$

41. $$(14t)^0$$

42. $$12a^0−15b^0$$

$$−3$$

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

43. $$6^{−2}$$

44. $$(−10)^{−3}$$

$$−\dfrac{1}{1000}$$

45. $$5·2^{−4}$$

46. $$(8n)^{−1}$$

$$\dfrac{1}{8n}$$

47. $$y^{−5}$$

48. $$10^{−3}$$

$$\dfrac{1}{1000}$$

49. $$\dfrac{1}{a^{−4}}$$

50. $$\dfrac{1}{6^{−2}}$$

$$36$$

51. $$−5^{−3}$$

52. $$\left(−\dfrac{1}{5}\right)^{−3}$$

$$−\dfrac{1}{25}$$

53. $$−(12)^{−3}$$

54. $$(−5)^{−3}$$

$$−\dfrac{1}{125}$$

55. $$\left(\dfrac{5}{9}\right)^{−2}$$

56. $$\left(−\dfrac{3}{x}\right)^{−3}$$

$$\dfrac{x^3}{27}$$

In the following exercises, simplify each expression using the Product Property.

57. $$(y^4)^3$$

58. $$(3^2)^5$$

$$3^{10}$$

59. $$(a^{10})^y$$

60. $$x^{−3}·x^9$$

$$x^5$$

61. $$r^{−5}·r^{−4}$$

62. $$(uv^{−3})(u^{−4}v^{−2})$$

$$\dfrac{1}{u^3v^5}$$

63. $$(m^5)^{−1}$$

64. $$p^5·p^{−2}·p^{−4}$$

$$\dfrac{1}{m^5}$$

In the following exercises, simplify each expression using the Power Property.

65. $$(k−2)^{−3}$$

66. $$\dfrac{q^4}{q^{20}}$$

$$\dfrac{1}{q^{16}}$$

67. $$\dfrac{b^8}{b^{−2}}$$

68. $$\dfrac{n^{−3}}{n^{−5}}$$

$$n^2$$

In the following exercises, simplify each expression using the Product to a Power Property.

69. $$(−5ab)^3$$

70. $$(−4pq)^0$$

$$1$$

71. $$(−6x^3)^{−2}$$

72. $$(3y^{−4})^2$$

$$\dfrac{9}{y^8}$$

In the following exercises, simplify each expression using the Quotient to a Power Property.

73. $$\left(\dfrac{3}{5x}\right)^{−2}$$

74. $$\left(\dfrac{3xy^2}{z}\right)^4$$

$$\dfrac{81x^4y^8}{z^4}$$

75. $$(4p−3q^2)^2$$

In the following exercises, simplify each expression by applying several properties.

76. $$(x^2y)^2(3xy^5)^3$$

$$27x^7y^{17}$$

77. $$(−3a^{−2})^4(2a^4)^2(−6a^2)^3$$

78. $$\left(\dfrac{3xy^3}{4x^4y^{−2}}\right)^2\left(\dfrac{6xy^4}{8x^3y^{−2}}\right)^{−1}$$

$$\dfrac{3y^4}{4x^4}$$

In the following exercises, write each number in scientific notation.

79. $$2.568$$

80. $$5,300,000$$

$$5.3×10^6$$

81. $$0.00814$$

In the following exercises, convert each number to decimal form.

82. $$2.9×10^4$$

$$29,000$$

83. $$3.75×10^{−1}$$

84. $$9.413×10^{−5}$$

$$0.00009413$$

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

85. $$(3×10^7)(2×10^{−4})$$

86. $$(1.5×10^{−3})(4.8×10^{−1})$$

$$0.00072$$

87. $$\dfrac{6×10^9}{2×10^{−1}}$$

88. $$\dfrac{9×10^{−3}}{1×10^{−6}}$$

$$9,000$$

### Multiply Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.

89. $$(−6p^4)(9p)$$

90. $$\left(\frac{1}{3}c^2\right)(30c^8)$$

$$10c^{10}$$

91. $$(8x^2y^5)(7xy^6)$$

92. $$\left(\frac{2}{3}m^3n^6\right)\left(\frac{1}{6}m^4n^4\right)$$

$$\dfrac{m^7n^{10}}{9}$$

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

93. $$7(10−x)$$

94. $$a^2(a^2−9a−36)$$

$$a^4−9a^3−36a^2$$

95. $$−5y(125y^3−1)$$

96. $$(4n−5)(2n^3)$$

$$8n^4−10n^3$$

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.

97. $$(a+5)(a+2)$$

98. $$(y−4)(y+12)$$

$$y^2+8y−48$$

99. $$(3x+1)(2x−7)$$

100. $$(6p−11)(3p−10)$$

$$18p^2−93p+110$$

In the following exercises, multiply the binomials. Use any method.

101. $$(n+8)(n+1)$$

102. $$(k+6)(k−9)$$

$$k^2−3k−54$$

103. $$(5u−3)(u+8)$$

104. $$(2y−9)(5y−7)$$

$$10y^2−59y+63$$

105. $$(p+4)(p+7)$$

106. $$(x−8)(x+9)$$

$$x^2+x−72$$

107. $$(3c+1)(9c−4)$$

108. $$(10a−1)(3a−3)$$

$$30a^2−33a+3$$

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. $$(5b−2)(3b^2+b−9)$$

$$15b^3−b^2−47b+18$$

In the following exercises, multiply. Use either method.

111. $$(m+6)(m^2−7m−30)$$

112. $$(4y−1)(6y^2−12y+5)$$

$$24y^2−54y^2+32y−5$$

Multiply Special Products

In the following exercises, square each binomial using the Binomial Squares Pattern.

113. $$(2x−y)^2$$

114. $$(x+\dfrac{3}{4})^2$$

$$x^2+\dfrac{3}{2}x+\dfrac{9}{16}$$

115. $$(8p^3−3)^2$$

116. $$(5p+7q)^2$$

$$25p^2+70pq+49q^2$$

In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

117. $$(3y+5)(3y−5)$$

118. $$(6x+y)(6x−y)$$

$$36x^2−y^2$$

119. $$(a+\dfrac{2}3b)(a−\dfrac{2}{3}b)$$

120. $$(12x^3−7y^2)(12x^3+7y^2)$$

$$144x^6−49y^4$$

121. $$(13a^2−8b4)(13a^2+8b^4)$$

### Divide Monomials

Divide Monomials

In the following exercises, divide the monomials.

122. $$72p^{12}÷8p^3$$

$$9p^9$$

123. $$−26a^8÷(2a^2)$$

124. $$\dfrac{45y^6}{−15y^{10}}$$

$$−3y^4$$

125. $$\dfrac{−30x^8}{−36x^9}$$

126. $$\dfrac{28a^9b}{7a^4b^3}$$

$$\dfrac{4a^5}{b^2}$$

127. $$\dfrac{11u^6v^3}{55u^2v^8}$$

128. $$\dfrac{(5m^9n^3)(8m^3n^2)}{(10mn^4)(m^2n^5)}$$

$$\dfrac{4m^9}{n^4}$$

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. $$(54y^4−24y^3)÷(−6y^2)$$

$$−9y^2+4y$$

131. $$\dfrac{63x^3y^2−99x^2y^3−45x^4y^3}{9x^2y^2}$$

132. $$\dfrac{12x^2+4x−3}{−4x}$$

$$−3x−1+\dfrac{3}{4x}$$

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

133. $$(4x^2−21x−18)÷(x−6)$$

134. $$(y^2+2y+18)÷(y+5)$$

$$y−3+\dfrac{33}{q+6}$$

135. $$(n^3−2n^2−6n+27)÷(n+3)$$

136. $$(a^3−1)÷(a+1)$$

$$a^2+a+1$$

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 $$x−3$$

$$2x^2−5x−4;\space0$$

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. $$\left(\dfrac{f}{g}\right)(−2)$$

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 $$x+2$$

$$−9$$

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 $$x^3−7x^2+7x−6$$

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?

a. trinomial b. 4

2. $$(5a^2+2a−12)(9a^2+8a−4)$$

3. $$(10x^2−3x+5)−(4x^2−6)$$

$$6x^2−3x+11$$

4. $$\left(−\dfrac{3}{4}\right)^3$$

5. $$x^{−3}x^4$$

$$x$$

6. $$5^65^8$$

7. $$(47a^{18}b^{23}c^5)^0$$

$$1$$

8. $$4^{−1}$$

9. $$(2y)^{−3}$$

$$\dfrac{1}{8y^3}$$

10. $$p^{−3}·p^{−8}$$

11. $$\dfrac{x^4}{x^{−5}}$$

$$x^9$$

12. $$(3x^{−3})^2$$

13. $$\dfrac{24r^3s}{6r^2s^7}$$

$$\dfrac{4r}{s^6}$$

14. $$(x4y9x−3)2$$

15. $$(8xy^3)(−6x^4y^6)$$

$$−48x^5y^9$$

16. $$4u(u^2−9u+1)$$

17. $$(m+3)(7m−2)$$

$$21m^2−19m−6$$

18. $$(n−8)(n^2−4n+11)$$

19. $$(4x−3)^2$$

$$16x^2−24x+9$$

20. $$(5x+2y)(5x−2y)$$

21. $$(15xy^3−35x^2y)÷5xy$$

$$3y^2−7x$$

22. $$(3x^3−10x^2+7x+10)÷(3x+2)$$

23. Use the Factor Theorem to determine if $$x+3$$ a factor of $$x^3+8x^2+21x+18$$.

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. $$(2.4×10^8)(2×10^{−5})$$

$$4.4×10^3$$

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. $$f(0)$$

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. $$\left(\dfrac{f}{g}\right)(3)$$

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 $$t=3$$.