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7.6: Chapter 5 Review Exercises - Mathematics


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)

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} ight)^{−3})

Answer

(−dfrac{1}{25})

53. (−(12)^{−3})

54. ((−5)^{−3})

Answer

(−dfrac{1}{125})

55. (left(dfrac{5}{9} ight)^{−2})

56. (left(−dfrac{3}{x} ight)^{−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} ight)^{−2})

74. (left(dfrac{3xy^2}{z} ight)^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}} ight)^2left(dfrac{6xy^4}{8x^3y^{−2}} ight)^{−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 ight)(30c^8))

Answer

(10c^{10})

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

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

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 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} ight)(x))
b. (left(dfrac{f}{g} ight)(−2))

Answer

a. (left(dfrac{f}{g} ight)(x)=x−6)
b. (left(dfrac{f}{g} ight)(−2)=−8)

141. For functions (f(x)=x^3+x^2−7x+2) and (g(x)=x−2), find a. (left(dfrac{f}{g} ight)(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} ight)^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. (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} ight)(x)) b. (left(dfrac{f}{g} ight)(3))

Answer

a. (left(dfrac{f}{g} ight)(x)=3x+4)
b. (left(dfrac{f}{g} ight)(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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