5.5: Review Exercise

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- 5.4E: Exercises
- 6: Polynomial and Rational Functions

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Chapter Review Exercises

Add and Subtract Polynomials

Determine the Degree of Polynomials

In the following exercises, determine the type of polynomial.

1. $16x^2−40x−25$

2. $5m+9$

Answer: binomial

3. $−15$

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

Answer: other polynomial

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

5. $4p+11p$

6. $−8y^3−5y^3$

Answer: $−13y^3$

7. $(4a^2+9a−11)+(6a^2−5a+10)$

8. $(8m^2+12m−5)−(2m^2−7m−1)$

Answer: $6m^2+19m−4$

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: $2x^2−2x+23$

In the following exercises, simplify.

13. $17mn^2−(−9mn^2)+3mn^2$

14. $18a−7b−21a$

Answer: $−7b−3a$

15. $2pq^2−5p−3q^2$

16. $(6a^2+7)+(2a^2−5a−9)$

Answer: $8a^2−5a−2$

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: $3y^2−8y−14$

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: $x^3+2x^2y−4xy^2$

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: 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: 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 $g(x)=8x^3−1$

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. $p^3·p^{10}$

Answer: $p^{13}$

31. $2·2^6$

32. $a·a^2·a^3$

Answer: $a^6$

33. $x·x^8$

34. $y^a·y^b$

Answer: $y^{a+b}$

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

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

Answer: $a^5$

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

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

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

39. $3^0$

40. $y^0$

Answer: $1$

41. $(14t)^0$

42. $12a^0−15b^0$

Answer: $−3$

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

43. $6^{−2}$

44. $(−10)^{−3}$

Answer: $−\dfrac{1}{1000}$

45. $5·2^{−4}$

46. $(8n)^{−1}$

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

47. $y^{−5}$

48. $10^{−3}$

Answer: $\dfrac{1}{1000}$

49. $\dfrac{1}{a^{−4}}$

50. $\dfrac{1}{6^{−2}}$

Answer: $36$

51. $−5^{−3}$

52. $\left(−\dfrac{1}{5}\right)^{−3}$

Answer: $−\dfrac{1}{25}$

53. $−(12)^{−3}$

54. $(−5)^{−3}$

Answer: $−\dfrac{1}{125}$

55. $\left(\dfrac{5}{9}\right)^{−2}$

56. $\left(−\dfrac{3}{x}\right)^{−3}$

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

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

57. $(y^4)^3$

58. $(3^2)^5$

Answer: $3^{10}$

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

60. $x^{−3}·x^9$

Answer: $x^5$

61. $r^{−5}·r^{−4}$

62. $(uv^{−3})(u^{−4}v^{−2})$

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

63. $(m^5)^{−1}$

64. $p^5·p^{−2}·p^{−4}$

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

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

65. $(k−2)^{−3}$

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

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

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.

69. $(−5ab)^3$

70. $(−4pq)^0$

Answer: $1$

71. $(−6x^3)^{−2}$

72. $(3y^{−4})^2$

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

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

75. $(4p−3q^2)^2$

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.

79. $2.568$

80. $5,300,000$

Answer: $5.3×10^6$

81. $0.00814$

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

82. $2.9×10^4$

Answer: $29,000$

83. $3.75×10^{−1}$

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 Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.

89. $(−6p^4)(9p)$

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

Answer: $10c^{10}$

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

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

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

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

93. $7(10−x)$

94. $a^2(a^2−9a−36)$

Answer: $a^4−9a^3−36a^2$

95. $−5y(125y^3−1)$

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.

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

98. $(y−4)(y+12)$

Answer: $y^2+8y−48$

99. $(3x+1)(2x−7)$

100. $(6p−11)(3p−10)$

Answer: $18p^2−93p+110$

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

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

102. $(k+6)(k−9)$

Answer: $k^2−3k−54$

103. $(5u−3)(u+8)$

104. $(2y−9)(5y−7)$

Answer: $10y^2−59y+63$

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

106. $(x−8)(x+9)$

Answer: $x^2+x−72$

107. $(3c+1)(9c−4)$

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.

113. $(2x−y)^2$

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

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

115. $(8p^3−3)^2$

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

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

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

117. $(3y+5)(3y−5)$

118. $(6x+y)(6x−y)$

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

Divide Monomials

In the following exercises, divide the monomials.

122. $72p^{12}÷8p^3$

Answer: $9p^9$

123. $−26a^8÷(2a^2)$

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: $x$

6. $5^65^8$

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

Answer: $1$

8. $4^{−1}$

9. $(2y)^{−3}$

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

10. $p^{−3}·p^{−8}$

11. $\dfrac{x^4}{x^{−5}}$

Answer: $x^9$

12. $(3x^{−3})^2$

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

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

14. $(x4y9x−3)2$

15. $(8xy^3)(−6x^4y^6)$

Answer: $−48x^5y^9$

16. $4u(u^2−9u+1)$

17. $(m+3)(7m−2)$

Answer: $21m^2−19m−6$

18. $(n−8)(n^2−4n+11)$

19. $(4x−3)^2$

Answer: $16x^2−24x+9$

20. $(5x+2y)(5x−2y)$

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 $t=3$ .

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Chapter Review Exercises

Add and Subtract Polynomials

Properties of Exponents and Scientific Notation

Multiply Polynomials

Divide Monomials

Chapter Practice Test

Chapter Review Exercises

Add and Subtract Polynomials

Properties of Exponents and Scientific Notation

Multiply Polynomials

Divide Monomials

Chapter Practice Test

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