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Algebra II - Polynomials ~1~ NJCTL.org Polynomial Functions NOTE: Some problems in this file are used with permission from the engageny.org website of the New York State Department of Education. Various files. Internet. Available from https://www.engageny.org/ccss-library. Accessed August, 2014. Properties of Exponents: Class Work Simplify the following expressions. 1. (−4 3 2 −2 ) −3 2. ( 4 3 3 2 ) 2 3. ( 3p 7 q 3 (2p 2 q 2 ) 3 ) −2 4. (5r 3 s 4 t 2 )(2r 3 s −3 ) 4 5. (3u 2 v −4 ) 3 (6u 4 v 3 ) −2 6. ( 8w 2 x −3 y 4 z 5 12w 3 x −4 y 5 z −6 ) −3 Properties of Exponents: Homework Simplify the following expressions. 7. (−3 −4 3 −3 ) −4 8. ( 4 4 6 3 −4 ) 2 9. ( 8p 7 q 9 (2p 2 q 2 ) 4 ) −2 10. 4(5r 10 s 12 t 8 )(2r 4 s −5 ) −3 11. (6u 6 v −3 ) 3 (9u 5 v −6 ) −2 12. ( 6w −3 x −4 y 5 z 6 15w 3 x −4 y 5 z −6 ) −2

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Page 1: Polynomial Functions - Center For Teaching & Learningcontent.njctl.org/courses/math/algebra-ii/polynomial-functions/... · Algebra II - Polynomials ~1~ NJCTL.org Polynomial Functions

Algebra II - Polynomials ~1~ NJCTL.org

Polynomial Functions

NOTE: Some problems in this file are used with permission from the engageny.org website of the

New York State Department of Education. Various files. Internet. Available from

https://www.engageny.org/ccss-library. Accessed August, 2014.

Properties of Exponents: Class Work

Simplify the following expressions.

1. (−4𝑔3ℎ2𝑗−2)−3

2. (4𝑘3

3𝑚𝑛2)2

3. (3p7q3

(2p2q2)3)−2

4. (5r3s4t2)(2r3s−3)4

5. (3u2v−4)3(6u4v3)−2

6. (8w2x−3y4z5

12w3x−4y5z−6)−3

Properties of Exponents: Homework

Simplify the following expressions.

7. (−3𝑔−4ℎ3𝑗−3)−4

8. (4𝑘4

6𝑚3𝑛−4)2

9. (8p7q9

(2p2q2)4)−2

10. 4(5r10s12t8)(2r4s−5)−3

11. (6u6v−3)3(9u5v−6)−2

12. (6w−3x−4y5z6

15w3x−4y5z−6)−2

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Algebra II - Polynomials ~2~ NJCTL.org

Operations with Polynomials: Class Work

Determine if each function is a polynomial function. If so, write it in standard form, name its degree, state its

type based on degree and based on number of terms, and identify the leading coefficient.

13. 2𝑥2 + 3𝑥2

14. 4

7𝑦 − 3𝑦2 + 3𝑦

15. 5𝑎3 − 2𝑎 − 4𝑎 + 3

16. 6𝑎2

𝑏− 5𝑎𝑏2 + 2𝑎𝑏2

17. (2𝑥−2 − 4) + (−5𝑥−2 − 3)

Perform the indicated operations.

18. (4g2 − 2) − (3g + 5) + (2g2 − g)

19. (6𝑡 − 3𝑡2 + 4) − (𝑡2 + 5𝑡 − 9)

20. (7𝑥5 + 8𝑥4 − 3𝑥) + (5𝑥4 + 2𝑥3 + 9𝑥 − 1)

21. (−10𝑥3 + 4𝑥2 − 5𝑥 + 9) − (2𝑥3 − 2𝑥2 + 𝑥 + 12)

22. The legs of an isosceles triangle are (3x2+ 4x +2) inches and the base is (4x-5) inches. Find the

perimeter of the triangle.

23. −2𝑎(4𝑎2𝑏 − 3𝑎𝑏2 − 6𝑎𝑏)

24. 7𝑗𝑘2(5𝑗3𝑘 + 9𝑗2 − 2𝑘 + 10)

25. (2x − 3)(4x + 2)

26. (𝑐2 − 3)(𝑐 + 4)

27. (m − 3)(2m2 + 4m − 5)

28. (2𝑓 + 5)(6𝑓2 − 4𝑓 + 1)

29. (3t2 − 2t + 9)(4t2 − t + 1)

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30. The width of a rectangle is (5x+2) inches and the length is (6x-7) inches. Find the area of the rectangle.

31. The radius of the base of a cylinder is (3x + 4) cm and the height is (7x + 2) cm. Find the volume of the

cylinder (V = 𝜋𝑟2ℎ).

32. A rectangle of (2x) ft by (3x-1) ft is cut out of a large rectangle of (4x+1)ft by (2x+2)ft. What is area of the shape that remains?

33. A pool that is 20ft by 30ft is going to have a deck of width x ft added all the way around the pool. Write an expression in simplified form for the area of the deck.

Multiply and simplify:

34. (𝑏 + 2)2

35. (𝑐 − 1)(𝑐 − 1)

36. (2𝑑 + 4𝑒)2

37. (5𝑓 + 9)(5𝑓 − 9)

38. What is the area of a square with sides (3x+2) inches? Expand, using the Binomial Theorem:

39. (2𝑥 + 4𝑦)5

40. (7𝑎 + 𝑏)3

41. (3𝑥 − 4𝑧)6

42. (𝑦 − 5𝑧)4

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Operations with Polynomials: Homework

Determine if each function is a polynomial function. If so, write it in standard form, name its degree, state its

type based on degree and based on number of terms, and identify the leading coefficient.

43. √2𝑥2 + 0.4𝑥3

44. 4

7𝑦− 8𝑦2 + 9𝑦

45. 11𝑎4 − 2𝑎3 + 7𝑎2 − 8𝑎 + 9

46. 6𝑎2

11−

5𝑎

9+ 2

47. (2𝑥2

3 − 4) + (−5𝑥2 − 3)

Perform the indicated operations:

48. (3n − 13) − (2n2 + 4n − 6) − (5𝑛 − 4)

49. (5g2 − 4) − (3g3 + 7) + (5g2 − 5g)

50. (−8𝑥4 + 7𝑥3 − 3𝑥 + 5) + (5𝑥4 + 2𝑥2 − 16𝑥 − 21)

51. (17𝑥3 − 9𝑥2 + 5𝑥 − 18) − (11𝑥3 − 2𝑥2 − 19𝑥 + 15)

52. The width of a rectangle is (5x2+6x +2) inches and the length is (6x-7) inches. Find the perimeter of the

rectangle.

53. 4𝑥(3𝑥2 − 5𝑥 − 2)

54. −6𝑎(3𝑎2𝑏 − 5𝑎𝑏2 − 7𝑏)

55. 8𝑗2𝑘3(2𝑗3𝑘 + 6𝑗2 − 5𝑘 + 11)

56. (4x + 5)(6x + 1)

57. (2𝑏 − 9)(4𝑏 − 2)

58. (2𝑐2 − 4)(3𝑐 + 2)

59. (2m − 5)(3m2 − 6m − 4)

60. (3𝑓 + 4)(6𝑓2 − 4𝑓 + 1)

61. (2𝑝2 − 5)(𝑝2 + 8𝑝 + 2)

62. (5t2 − 3t + 6)(3t2 − 2t + 1)

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63. The width of a rectangle is (4x-3) inches and the length is (3x-5) inches. Find the area of the rectangle.

64. The radius of the base of a cone is (9x - 3) cm and the height is (3x + 2) cm. Find the volume of the

cylinder (V = 1

3𝜋𝑟2ℎ).

65. A rectangle of (3x) ft by (5x-1) ft is cut out of a large rectangle of (6x+2)ft by (3x+4)ft. What is area of the

shape that remains?

66. A pool that is 25ft by 40ft is going to have a deck of width (x + 2) ft added all the way around the pool.

Write an expression in simplified form for the area of the deck.

Multiply and simplify: 67. (3𝑎 − 1)(3𝑎 + 1)

68. (𝑏 − 2)2

69. (𝑐 − 1)(𝑐 + 1)

70. (3𝑑 − 5𝑒)2

71. (5𝑓 + 9)(5𝑓 + 9)

72. What is the area of a square with sides of (4x-6y) inches?

Expand the following using the binomial Theorem:

73. (2𝑎 − 𝑏)6

74. (3𝑥 + 2𝑦)3

75. (5𝑦 − 4𝑧)5

76. (𝑎 + 7𝑏)4

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Factoring I Classwork

Factoring out the GCF

77. 6x3y2 – 3x2y

78. 10p3q – 15p3q2 – 5p2q2

79. 7m3n3 – 7m3n2 + 14m3

Factoring ax2 + bx + c

80. x2 – 5x – 24

81. m2 – mn – 6n2

82. x2 – 2xy + y2

83. a2 + ab – 12b2

84. x2 – 6xy + 8y2

85. 2x2 + 7x + 3

86. 6x2 – x – 2

87. 5a2 + 17a – 12

88. 6m2 - 5mn + n2

89. 6p2 + 37p + 6

90. 4c2 + 20cd + 25d2

Factoring I Homework

Factoring out the GCF

91. 8x3y – 4x2y2

92. 8m3n3 – 4m2n3 – 32mn3

93. -18p3q2 + 3pq

Factoring ax2 + bx + c

94. m2 – 2m – 24

95. a2 – 13a + 12

96. n2 + n – 6

97. x2 – 10xy + 21y2

98. x2 + 11xy + 18y2

99. 6x2 – 5x + 1

100. 15p2 – 22p – 5

101. 10m2 + 13m – 3

102. 12x2 – 7xy + y2

103. 4p2 + 24p + 35

104. 15m2 – 13mn + 2n2

Spiral Review

105. Simplify: 106. Multiply: 107. Divide 108. Evaluate, use x = 5:

5 – 4 [(-2) – (-2)] 23

4 ∙ 4

2

3 2

3

4 ÷ 4

2

3 -2(-6x – 9) + 4

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Factoring II Classwork

Factoring a2 – b2, a3 – b3, a3 + b3

109. a3 – 1

110. 25x2 – 16y2

111. 121a2 – 16b2

112. 27x3 + 8y3

113. a3b3 – c3

114. 4x2y2 – 1

Factoring by Grouping

115. 2xy + 5x + 8y + 20

116. 9mn – 3m – 15n + 5

117. 2xy – 10x – 3y + 15

118. 10rs – 25r + 6s – 15

119. 10pq – 2p – 5q + 1

120. 10mn + 5m + 6n + 3

Mixed Factoring

121. 3x3 – 12x2 + 36x

122. 6m3 + 4m2 – 2m

123. 3a3b – 48ab

124. 54x4 + 2xy3

125. x4y + 12x3y + 20x2y

Factoring II Homework

Factoring a2 – b2, a3 – b3, a3 + b3

126. y3 + 27

127. 64m3 – 1

128. p2 – 36q2

129. m2n2 – 4

130. x2 + 16

131. 8x3 – 27y3

Factoring by Grouping

132. 6mp – 2m – 15p + 5

133. 6xy + 15x + 4y + 10

134. 4rs – 4r + 3s – 3

135. 6tr – 9t – 2r + 3

136. 8mn + 4m + 6n + 3

137. 3xy – 4x – 15y + 20

Mixed Factoring

138. 3m3 – 3mn2

139. -6x3 – 28x2 + 10x

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140. 18a3b – 50ab

141. x4y + 27xy

142. -12r3 – 21r2 – 9r

143. 2x2y2 – 2x2y – 2xy2 + 2xy

Spiral Review

144. Simplify: 145. Simplify: 146. Add: 147. Evaluate, use x = -3, y = 2

8(-4) (2)(-1) + (4)2 172 - (12 - 4)2 + 2 22

7+ 5

3

5 -3x + 2y – xy + x

Division of Polynomials: Class Work

Simplify.

148. 6x3−3x2+9x

3x

149. (4𝑎4𝑏3 + 8𝑎3𝑏3 − 6𝑎2𝑏2) ÷ (2𝑎2𝑏)

150. 6x3−4x2+7x+3

3x+1

151. (4𝑎4 + 8𝑎3 − 6𝑎2 + 3𝑎 + 4) ÷ (𝑎 − 1)

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152. Consider the polynomial function 𝒇(𝒙) = 𝟑𝒙𝟐 + 𝟖𝒙 − 𝟒.

a. Divide 𝒇 by 𝒙 − 𝟐. b. Find 𝒇(𝟐).

153. Consider the polynomial function 𝒈(𝒙) = 𝒙𝟑 − 𝟑𝒙𝟐 + 𝟔𝒙 + 𝟖.

a. Divide 𝒈 by 𝒙 + 𝟏.

b. Find 𝒈(−𝟏).

154. Consider the polynomial 𝑃(𝑥) = 𝑥3 + 𝑥2 − 10𝑥 − 10.

Is 𝑥 + 1 one of the factors of 𝑃? Explain.

155. The volume a hexagonal prism is (3𝑡3 − 4𝑡2 + 𝑡 + 2) 𝑐𝑚3 and its height is (t+1) cm. Find the area of

the base. (Use V=Bh)

Division of Polynomials: Homework

Simplify.

156. 16x5−12x3+24x2

4x2

157. (4𝑎4𝑏3 + 8𝑎3𝑏3 − 16𝑎2𝑏2) ÷ (4𝑎𝑏2)

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158. (3f 3 + 18f − 12)(3f 2)−1

159. 3x3−3x2+9x+2

x+3

160. Consider the polynomial function 𝒇(𝒙) = 𝒙𝟑 − 𝟐𝟒.

a. Divide 𝒇 by 𝒙 − 𝟐. b. Find 𝒇(𝟐).

161. Consider the polynomial function 𝒈(𝒙) = 𝒙𝟑 + 𝟓𝒙𝟐 − 𝟖𝒙 + 𝟕.

b. Divide 𝒈 by 𝒙 + 𝟏.

c. Find 𝒈(−𝟏).

162. Consider the polynomial 𝑃(𝑥) = 2𝑥3 + 5𝑥2 − 12𝑥 + 5.

Is 𝑥 − 1 one of the factors of 𝑃? Explain.

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163. (8f 3)(2f + 4)−1

164. The volume a hexagonal prism is (4𝑡3 − 3𝑡2 + 2𝑡 + 2) 𝑐𝑚3 . The area of the base, B is (t-1) cm2. Find

the height of the prism. (Use V=Bh)

165. Consider the polynomial 𝑃(𝑥) = 𝑥4 + 3𝑥3 − 28𝑥2 − 36𝑥 + 144.

a. Is 1 a zero of the polynomial 𝑃?

b. Is 𝑥 + 3 one of the factors of 𝑃?

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Characteristics of Polynomial Functions: Class Work

For each function or graph answer the following questions:

a. Does the function have even degree or odd degree?

b. Is the lead coefficient positive or negative?

c. Is the function even, odd or neither?

166. 167.

168. 169.

Is each function below odd, even or neither?

170. 𝑓(𝑥) = 2𝑥4 + 3𝑥2 − 2

171. 𝑦 = 5𝑥5 − 3𝑥 + 1

172. 𝑔(𝑥) = −2𝑥(4𝑥2 − 3𝑥)

173. ℎ(𝑥) = 4𝑥

174. For each function in #’s 170 – 173 above, describe the end behavior in these terms: as x∞,

f(x) ____, and as x -∞, f(x) _____.

Is each function below odd, even or neither? How many zeros does each function appear to have?

175. 176. 177. 178.

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Algebra II - Polynomials ~13~ NJCTL.org

Characteristics of Polynomial Functions: Homework

For each function or graph answer the following questions:

a. Does the function have even degree or odd degree?

b. Is the lead coefficient positive or negative?

c. Is the function even, odd or neither?

179. 180.

181. 182.

Is each function below an odd-function, an even-function or neither.

183. 𝑓(𝑥) = 5𝑥4 − 6𝑥2 + 3𝑥

184. 𝑦 = 5𝑥5 − 3𝑥3 + 1𝑥

185. 𝑔(𝑥) = 2𝑥2(4𝑥3 − 3𝑥)

186. ℎ(𝑥) = −4

5𝑥2 + 2

187. For each function in #’s 183 – 186 above, describe the end behavior in these terms: as x∞, f(x)

____, and as x -∞, f(x) _____.

Are the following functions odd, even or neither? How many zeros does the function appear to have?

188. 189. 190. 191.

Analyzing Graphs and Tables of Polynomial Functions: Class Work

Identify any zeros (either as an integer or as an interval of x-values) of the function. Label any relative

maximum and minimum.

192. 193.

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194. 195.

196. 197. 198.

Analyzing Graphs and Tables of Polynomial Functions: Homework

Identify any zeros (either as an integer or as an interval of x-values) of the function. Label any relative

maximum and minimum.

199. 200.

201. 202.

203. 204. 205.

x f(x)

-2 5

-1 1

0 -1

1 0

2 2

3 1

4 -1

x f(x)

-2 2

-1 -3

0 -4

1 -1

2 2

3 5

4 -2

x f(x)

-2 -4

-1 0

0 2

1 1

2 -1

3 -3

4 -1

x f(x)

-2 2

-1 4

0 2

1 -2

2 0

3 3

4 1

x f(x)

-2 6

-1 2

0 1

1 3

2 1

3 -1

4 0

x f(x)

-2 4

-1 -2

0 -3

1 -1

2 1

3 3

4 7

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Zeros and Roots of a Polynomial Function: Class Work

For each graph below and its given degree, name the real zeros and their multiplicity, and state the number of

imaginary zeros.

206. 207. 208.

4th degree 4th degree 5th degree

Name all of the real and imaginary zeros and state their multiplicity.

209. 𝑓(𝑥) = (𝑥 + 1)(𝑥 + 2)(𝑥 + 2)(𝑥 − 3)

210. 𝑔(𝑥) = (𝑥2 − 1)(𝑥2 + 1)

211. 𝑦 = (𝑥 + 1)2(𝑥 + 2)(𝑥 − 2)

212. ℎ(𝑥) = 𝑥2(𝑥 − 10)(𝑥 + 1)

213. 𝑦 = (𝑥2 − 9)(𝑥 + 3)2(𝑥2 + 9)

Zeros and Roots of a Polynomial Function: Homework

For each graph below and its given degree, name the real zeros and their multiplicity, and state the number of

imaginary zeros.

214. 215. 216.

3rd degree 4th degree 6th degree

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Name all of the real and imaginary zeros and state their multiplicity.

217. 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 3)(𝑥 + 3)(𝑥 − 3)

218. 𝑔(𝑥) = (𝑥2 − 4)(𝑥2 + 4)

219. 𝑦 = (𝑥 + 7)2(4𝑥2 − 64)

220. ℎ(𝑥) = 𝑥3(𝑥 − 7)(𝑥 − 6)𝑥(2𝑥 + 4)(𝑥 − 5)

221. 𝑦 = (𝑥 + 4)2(𝑥2 − 16)(𝑥2 + 16)

Zeros and Roots of a Polynomial Function by Factoring: Class Work

Name all of the real and imaginary zeros and state their multiplicity.

222. 𝑓(𝑥) = 2𝑥3 + 16𝑥2 + 30𝑥 225. 𝑓(𝑥) = 𝑥4 − 8𝑥2 − 9

223. 𝑓(𝑥) = 𝑥4 + 9𝑥2 226. 𝑓(𝑥) = 2𝑥3 + 𝑥2 − 16𝑥 − 15

224. 𝑓(𝑥) = 2𝑥3 + 3𝑥2 − 8𝑥 − 12 227. 𝑓(𝑥) = 𝑥3 + 4𝑥2 − 25𝑥 − 100

228. Consider the function 𝑓(𝑥) = 𝑥3 + 3𝑥2 − 𝑥 − 3.

a. Use the fact that 𝑥 + 3 is a factor of 𝑓 to factor this polynomial.

b. Find the x-intercepts for the graph of 𝑓.

c. At which 𝒙-values can the function change from being positive to negative or from negative to

positive?

d. For 𝒙 < −𝟑, is the graph above or below the 𝒙-axis? How can you tell?

e. For −𝟑 < 𝒙 < −𝟏, is the graph above or below the 𝒙-axis? How can you tell?

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f. For −𝟏 < 𝒙 < 𝟏, is the graph above or below the 𝒙-axis? How can you tell?

g. For 𝒙 > 𝟏, is the graph above or below the 𝒙-axis? How can you tell?

h. Use the information generated in parts (f)–(i) to sketch a graph of 𝒇.

Zeros and Roots of a Polynomial Function by Factoring: Homework

Name all of the real and imaginary zeros and state their multiplicity.

229. 𝑓(𝑥) = 𝑥3 − 3𝑥2 − 2𝑥 + 6 232. 𝑓(𝑥) = 𝑥4 − 𝑥2 − 30

230. 𝑓(𝑥) = 𝑥4 + 𝑥2 − 12 233. 𝑓(𝑥) = 3𝑥4 − 5𝑥3 + 𝑥2 − 5𝑥 − 2

231. 𝑓(𝑥) = 𝑥3 + 5𝑥2 − 9𝑥 − 45 234. 𝑓(𝑥) = 𝑥4 − 5𝑥3 + 20𝑥 − 16

235. Consider the function 𝑓(𝑥) = 𝑥3 − 6𝑥2 − 9𝑥 + 14.

a. Use the fact that 𝑥 + 2 is a factor of 𝑓 to factor this polynomial.

b. Find the x-intercepts for the graph of 𝑓.

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c. At which 𝒙-values can the function change from being positive to negative or from negative to

positive?

d. For 𝒙 < −𝟐, is the graph above or below the 𝒙-axis? How can you tell?

e. For −𝟐 < 𝒙 < 𝟏, is the graph above or below the 𝒙-axis? How can you tell?

f. For 𝟏 < 𝒙 < 𝟕, is the graph above or below the 𝒙-axis? How can you tell?

g. For 𝒙 > 𝟕, is the graph above or below the 𝒙-axis? How can you tell?

h. Use the information generated in parts (f)–(i) to sketch a graph of 𝒇.

Writing Polynomials from Given Zeros: Class work

Write a polynomial function of least degree with integral coefficients that has the given zeros.

236. −3, −2, 2 240.

237. −3, −1, 2, 4

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238. ±√3,1

3, −5 241.

239. 2, 3, 𝑖, −𝑖,3

5

Writing Polynomials from Given Zeros: Homework

Write a polynomial function of least degree with integral coefficients that has the given zeros.

242. 1, 2,3

4 246.

243. −1, 3, 0

244. 0 (𝑚𝑢𝑙𝑡. 2), −5, 1

245. −2𝑖, 2𝑖, −5(𝑚𝑢𝑙𝑡. 3) 247.

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UNIT REVIEW

Multiple Choice

1. Simplify the following expression: (6p8q9

(2p3q4)3)−2

a. 3

4pq3

b. 9

16p2q6

c. 4pq3

3

d. 16p2q6

9

2. The sides of a rectangle are (2x2 – 11x +1) ft and (3x – 4) ft find the perimeter of the rectangle.

a. (2x2 – 8x – 3) ft

b. (4x2 – 16x – 6)

c. (5x3 – 11x – 3) ft

d. (6x3 – 41x2 + 47x – 4) ft2

3. The sides of a rectangle are (2x2 – 11x +1) ft and (3x – 4) ft find the area of the rectangle.

a. (6x3 – 41x2 – 41x – 4) ft2

b. (6x3 – 25x2 + 47x – 4) ft2

c. (6x3 – 41x2 + 47x – 4) ft2

d. (6x3 – 33x – 4) ft2

4. A pool that is 10ft by 20 ft is going to have a deck (x) ft added all the way around the pool. Write an

expression in simplified form for the area of the deck.

a. (60x + 4x2)ft2

b. (30x + x2)ft2

c. (200 + 60x + 4x2)ft2

d. (200 + 30x + x2)ft2

5. What is the area of a square with sides (6x – 2) inches?

a. (36x2 − 4) in2

b. (36x2 + 4) in2

c. (36x2 − 12x − 4) in2

d. (36x2 − 24x + 4) in2

6. 27w3x5−12w4x3+24w3x2

6w2x2 is equivalent to which of the following?

a. 9wx3−4w2x+4w

3

b. 9wx3

2− 2w2x + 4w

c. 9wx3−4w2x

3+ 4w

d. 9wx3+4w2x+8w

2

7. (2𝑎4 − 6𝑎2 + 4) ÷ (𝑎 − 2)

a. 2𝑎3 − 3𝑎 − 2

b. 2𝑎3 − 3𝑎2 − 2

c. 2𝑎3 + 4𝑎2 − 2𝑎 − 4 +−4

𝑎−2

d. 2𝑎3 + 4𝑎2 + 2𝑎 + 4 +12

𝑎−2

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8. A box has volume of (3x2 − 2x − 5) cm3 and a height of (x+1) cm. Find the area of the base of the box.

a. (3x + 2) cm2

b. (3x – 2) cm2

c. (3x + 5) cm2

d. (3x – 5) cm2

9. Using the graph, decide if the following function has an odd or even degree and the sign of the lead

coefficient.

a. odd degree; positive

b. odd degree; negative

c. even degree; positive

d. even degree; negative

10. Which of the following equations is of an odd-function?

a. 𝑦 = 3𝑥5 − 2𝑥

b. 𝑦 = 5𝑥7 − 3𝑥3 + 9

c. 𝑦 = 𝑥5(𝑥7 + 𝑥5)

d. 𝑦 = 7𝑥10

11. What value should A be in the table so that the function has 4 zeros?

a. -2

b. 0

c. 1

d. 3

12. Name all of the real and imaginary zeros and state their multiplicity:

𝑦 = (𝑥2 + 8𝑥 + 16)(4𝑥2 + 64)

a. Real zeros: -4 with multiplicity 2; Imaginary zeros: ± 4i each with multiplicity 1

b. Real zeros: -4 with multiplicity 3, 4 with multiplicity 1; No imaginary zeros

c. Real zeros: -4 with multiplicity 4; No imaginary zeros

d. Real zeros: -4 with multiplicity 2; Imaginary zeros: 2i with multiplicity 2

Extended Response

1. Graph 𝑦 = (𝑥 + 2)2(𝑥 + 1)𝑥(𝑥 − 1)(𝑥 − 3).

Name the real zeros and their multiplicity.

x f(x)

-2 6

-1 A

0 2

1 3

2 1

3 -1

4 0

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2. Given the function 𝑓(𝑥) = 3𝑥3 + 3𝑥2 − 6. Write the function in factored form.

3. Name all of the real and imaginary zeros and state their multiplicity of the function

𝑓(𝑥) = 𝑥3 − 10𝑥2 + 11𝑥 + 70

4. Write a polynomial function of least degree with integral coefficients that has the given zeros.

-4.5, -1, 0, 1, 4.5

5. Consider the graph of a degree 5 polynomial shown to the

right, with 𝑥-intercepts −4, −2, 1, 3, and 5.

Write an equation for a possible polynomial function that

the graph represents.

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Answer Key

1. 𝑗6

−64𝑔9ℎ6

2. 16𝑘6

9𝑚2𝑛4

3. 64𝑞6

9𝑝2

4. 80𝑟15𝑡2

𝑠8

5. 3

4𝑢2𝑣18

6. 27𝑤3𝑦3

8𝑥3𝑧33

7. 𝑔16𝑗12

81ℎ12

8. 4𝑘8𝑛8

9𝑚6

9. 4𝑝2

𝑞2

10. 5𝑠27𝑡8

2𝑟2

11. 8𝑢8𝑣3

3

12. 25𝑤12

4𝑧24

13. Yes, 5x2, degree: 2, monomial/quadratic,

5

14. Yes, -3y2+34

7y, degree: 2,

binomial/quadratic, -3

15. Yes, 5a3-6a+3, degree: 3,

trinomial/cubic, 5

16. Not a polynomial function

17. Not a polynomial function

18. 6g2-4g-7

19. -4t2+t+13

20. 7𝑥5 + 13𝑥4 + 2𝑥3 + 6𝑥 −1

21. −12𝑥3 + 6𝑥2 − 6𝑥 − 3

22. Perimeter = (6x2+12x-1) inches

23. -8a3b+6a2b2+12a2b

24. 35j4k3+63j3k2-14jk3+70jk2

25. 8x2-8x-6

26. c3+4c2-3c-12

27. 2m3-2m2-17m+15

28. 12f3+22f2-18f+5

29. 12t4-11t3+41t2-11t+9

30. Area = (30x2-23x-14) in.2

31. Area = 𝜋(63x3+186x2+160x+32) m2

32. Area = (2x2+12x+2) ft.2

33. Areadeck = (4x2+100x) ft.2

34. b2+4b+4

35. c2-2c+1

36. 4d2+16de+16e2

37. 25f2-81

38. (9x2+12x+4) in.2 39. 32x5+320x4y+1280x3y2+2560x2y3+2560xy4+1024y5

40. 343a3+147a2b+21ab2+b3 41. 729x6-5832x5z+19440x4z2-34560x3z3+34560x2z4-

18432xz5+4096z6

42. y4-20y3z+150y2z2-500yz3+625z4

43. Yes, 0.4𝑥3 + √2𝑥2, degree: 3,

binomial/cubic, 0.4

44. Not a polynomial function

45. Yes, already in std form, degree: 4, no

specific name/quartic, 11

46. Yes, already in std form, degree: 2,

trinomial/quadratic, 6/11

47. Not a polynomial function

48. -2n2-6n-3

49. -3g3+10g2-5g-11

50. -3x4+7x3+2x2-19x-16

51. 6x3-7x2+24x-33

52. Perimeter = (10x2+24x-10) inches

53. 12x3-20x2-8x

54. -18a3b+30a2b2+42ab

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55. 16j5k4+48j4k3-40j2k4+88j2k3

56. 24x2+34x+5

57. 8b2-40b+18

58. 6c3+4c2-12c-8

59. 6m3-27m2+22m+20

60. 18f3+12f2-13f+4

61. 2p4+16p3-p2-40p-10

62. 15t4-19t3+29t2-15t+6

63. Area = (12x2-29x+15) in.2

64. Area = 81x3-27x+6) m2

65. Area = (3x2+33x+8) in.2

66. Areadeck = (4x2+146x+276) ft.2

67. 9a2-1

68. b2-4b+4

69. c2-1

70. 9d2-30de+25e2

71. 25f2+90f+81

72. Area = (16x2-48xy+36y2) in.2 73. 64a6-192a5b+240a4b2-160a3b3+60a2b4-12ab5+b6

74. 27x3+54x2y+36xy2+8y3 75. 3125y5-12500y4z+20000y3z2-16000y2z3+6400yz4-1024z5

76. a4+28a3b+294a2b2+1372ab3+2401b4

77. 3x2y(2xy – 1)

78. 5p2q(2p – 3pq – q)

79. 7m3(n3 – n2 + 2)

80. (x – 8)(x + 3)

81. (m – 3n)(m + 2n)

82. (x – y)(x – y)

83. (a + 4b)(a – 3b)

84. (x – 4y)(x – 2y)

85. (2x + 1)(x + 3)

86. (3x – 2)(2x + 1)

87. (5a – 3)(a + 4)

88. (2m – n)(3m – n)

89. (6p + 1)(p + 6)

90. (2c + 5d)(2c + 5d)

91. 4x2y(2x-y)

92. 4mn3(2m2-m-8)

93. 3pq(-6p2q+1)

94. (m - 6)(m + 4)

95. (a - 12)(a - 1)

96. (n + 3)(n - 2)

97. (x – 7y)(x – 3y)

98. (x + 9y)(x + 2y)

99. (3x – 1)(2x – 1)

100. (3p – 5)(5p + 1)

101. (2m + 3)(5m – 1)

102. (3x – y)(4x – y)

103. (2p + 7)(2p + 5)

104. (3m – 2n)(5m – n)

105. 5

106. 77

6

107. 33

56

108. 82

109. (a – 1)(a2 + a + 1)

110. (5x – 4y)(5x + 4y)

111. (11a – 4b)(11a + 4b)

112. (3x + 2y)(9x2 + 6xy + 4y2)

113. (ab – c)(a2b2 + abc + c2)

114. (2xy – 1)(2xy + 1)

115. (x + 4)(2y + 5)

116. (3m – 5)(3n – 1)

117. (2x – 3)(y – 5)

118. (5r + 3)(2s – 5)

119. (2p – 1)(5q – 1)

120. (5m + 3)(2n + 1)

121. 3x(x – 6)(x + 2)

122. 2m(3m – 1)(m + 1)

123. 3ab(a – 4)(a + 4)

124. 2x(3x + y)(9x2 – 3xy + y2)

125. x2y(x + 10)(x + 2)

126. (y + 3)(y2 – 3y + 9)

127. (4m – 1)(16m2 + 4m + 1)

128. (p – 6q)(p + 6q)

129. (mn – 2)(mn + 2)

130. Not Factorable

131. (2x – 3y)(4x2 + 6xy + 9y2)

132. (2m – 5)(3p – 1)

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133. (3x + 2)(2y + 5)

134. (4r + 3)(s – 1)

135. (3t – 1)(2r – 3)

136. (4m + 3)(2n + 1)

137. (x – 5)(3y – 4)

138. 3m(m – n)(m + n)

139. -2x(3x – 1)(x + 5)

140. 2ab(3a – 5)(3a + 5)

141. xy(x + 3)(x2 – 3x + 9)

142. -3r(4r + 3)(r + 1)

143. 2xy(x – 1)(y – 1)

144. 32

145. 227

146. 731

35

147. 16

148. 2x2-x+3

149. 2a2b2+4ab2-3b

150. 2x2 – 2x + 3

151. 4a3+12a2+6a+9 + 13

𝑎−1

152. a. 3𝑥 + 14 +24

𝑥−2 b. 24

153. a. 𝑥2 − 4𝑥 + 10 −2

𝑥+1 b. -2

154. Yes, because P(-1) = 0.

155. B = (3t2-7t + 8 - 6

𝑡+1) cm.2

156. 4x3-3x+6

157. a3b+2a2b- 4a

158. f+ 6

𝑓 -

4

𝑓2

159. 3x2-12x+45 - 133

𝑥+3

160. a. 𝑥2 + 2𝑥 + 4 −16

𝑥−2 b. -16

161. a. 𝑥2 + 4𝑥 − 12 +19

𝑥+1 b. 19

162. Yes, because P(1) = 0.

163. 4f2-8f+16 - 32

𝑓+2

164. height = 4t2+t+3+ 5

𝑡−1 cm

165. a. No b. Yes

166. Odd; positive; neither

167. Even; negative; even

168. Even; positive; neither

169. Odd; negative; neither

170. Even function

171. Neither

172. Neither

173. Odd

174. 170: ∞, ∞ 171: ∞, −∞ 172: −∞, ∞

173: ∞, −∞

175. Odd function; 3 zeros

176. Even function; 2 zeros

177. Neither; 3 zeros

178. Even function; 2 zeros

179. Odd; negative; neither

180. Even; negative; even

181. Even; positive; even

182. Odd; negative; odd

183. Neither

184. Odd function

185. Odd function

186. Even function

187. 184: ∞, ∞ 185: ∞, −∞ 186: ∞, −∞

187: −∞, − ∞

188. Even function; 2 zeros

189. Odd function; 1 zero

190. Neither; 2 zeros

191. Odd function; 1 zero

192. Zeros: between x= -2 and x= -1, at x= 0,

between x=1 and x= 2; relative max at x=

-1; relative min at x=1

193. Zeros: between x=-2 and x=-1,

between x=-1 and x=0, between x=0 and

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x=1, between x=1 and x=2; relative max at

x=-1 and x=1; relative min at x=0

194. Zeros: at x=-2 and x=2; no relative max;

relative min at x=0

195. Zeros: between x=-2 and x=-1,

between x=-1 and x=0 , at x=0, between

x=0 and x=1, between x=1 and 2; relative

max at x≈-.5 and x≈1.5; relative min at

x≈-1.5 and x≈.5

196. Zeros: between x=-1 and 0, at x=1,

between x=3 and 4; relative max x=2;

relative min at x=0

197. Zeros: at x=-1, between x=1 and 2;

relative max at x=0; relative min at x=3

198. Zeros: between x=-2 and x=-1,

between x=1 and x=2, between x=3 and

x=4; relative max at x=3; relative min at

x=0

199. Zero: at x=2; no relative max or min

200. Zeros: at x≈-2, x≈-1, x≈0, x≈1,and

x≈2; relative max at x=-1.5 and x=.5;

relative min at x=-.5 and x=1.5

201. Zeros: between x=-2 and x=-1,

between x=1 and x=2; relative max at x=0;

relative min at x=-1 and x=1

202. No zeros; relative max at x=0; relative

min at x=-1 and x=1

203. Zeros: between x=2 and 3, and at x=4;

relative max at x=1; relative min at x=0

and x=3

204. Zeros: between x=0 and 1, at x=2;

relative max at x=-1 and x=3; relative min

at x=1

205. Zeros: between x=-2 and x=-1,

between x=1 and x=2; no relative max;

relative min at x=0

206. Real zeros: at x=-2 and x=2 ( both mult.

of 2); no imaginary zeros

207. Real zeros: at x=3 (mult. of 2); 2

imaginary zeros

208. Real zeros: at x= −3, x = -1, x=3 (all

mult. of 1), x=3 (mult. of 2); no imaginary

zeros

209. Real zeros: at x=-1 (mult. of 1), at x=-2

(mult. of 2) and x=3 (mult. of 1)

210. Real zeros: at x=-1 (mult. of 1), at x=1

(mult. of 1); Imaginary zeros: at x= i (mult.

of 1), at x=-i (mult. of 1)

211. Real zeros: at x=-1 (mult. of 2), at x=2

(mult. of 1), at x=-2 (mult. of 1)

212. Real zeros: at x=0 (mult. of 2), at x=10

(mult. of 1), at x=-1 (mult. of 1)

213. Real zeros: at x=-3 (mult. of 3), at x=3

(mult. of 1); Imaginary zeros: at x=3i (mult.

of 1), at x=-3i (mult. of 1)

214. Real zeros: at x=-2 (mult. of 1) and at

x=-1 (mult. of 1) and at x = 1 (mult. of 1);

no imaginary zeros

215. Real zeros: at x=-2 and x=2 (each mult.

of 1); 2 imaginary zeros

216. Real zeros: at x=-1.5 (mult. of 1) x=2

(mult. of 1) and at x=3 (mult. of 2); 2

imaginary zeros

217. Real zeros: at x=1 (mult. of 1), at x=-3

(mult. of 2), at x=3 (mult. of 1)

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218. Real zeros: at x=2 (mult. of 1), at x=-2

(mult. of 1); Imaginary zeros: at x=2i (mult.

of 1), at x=-2i (mult. of 1)

219. Real zeros: at x=-7 (mult. of 2), x=4

(mult. of 1), at x=-4 (mult. of 1)

220. Real zeros: at x=0 (mult. of 4), at x=7

(mult. of 1), at x=6 (mult. of 1), at x=-2

(mult. of 1), at x=5 (mult. of 1)

221. Real zeros: at x=-4 (mult. of 3), at x=4

(mult. of 1); Imaginary zeros: at x=4i (mult.

of 1), at x=-4i (mult. of 1)

222. Real zeros: at x=0 (mult. of 1), at x=-3

(mult. of 1), at x=-5 (mult. of 1

223. Real zeros: at x=0 (mult. of 2) 2

Imaginary zeros: at x= 3i (mult. of 1), at

x=-3i (mult. of 1)

224. Real zeros: at x=-1.5 (mult. of 1), at x=

2 (mult. of 1), at x=-2 (mult. of 1)

225. Real zeros: at x=-3 (mult. of 1), at x=3

(mult. of 1);

2 Imaginary zeros: at x= i (mult. of 1),

at x=-i (mult. of 1)

226. Real zeros: at x=-1 (mult. of 1), at

x=-5

2 (mult. of 1), at x=3 (mult. of 1)

227. Real zeros: at x=-5 (mult. of 1), at x=-4 (mult. of 1), at x=5 (mult. of 1)

228. a. f(x) = (x + 3)(x + 1)(x – 1)

b. -3, -1, 1

c. -3, -1, 1

d. Below, f(-4) is negative, OR since the

degree is 3 and the leading coefficient is

positive.

e. Above, crosses at -3

f. Below, crosses at -1

g. Above, crosses at 1

h.

229. 3 Real zeros: at 𝑥 = √2 (mult. of 1), at

𝑥 = −√2 (mult. of 1), at x=3 (mult. of 1)

230. Real zeros: at 𝑥 = √3 (mult. of 1), at

𝑥 = −√3 (mult. of 1);

2 Imaginary zeros: at 𝑥 = 2𝑖 (mult. of

1), at 𝑥 = −2𝑖 (mult. of 1)

231. Real zeros: at x=-3 (mult. of 1), at x= 3

(mult. of 1), at x=-5 (mult. of 1)

232. 2 Real zeros: at 𝑥 = √6 (mult. of 1), at

𝑥 = −√6 (mult. of 1);

2 Imaginary zeros: at 𝑥 = 𝑖√5 (mult. of

1), at 𝑥 = −𝑖√5 (mult. of 1)

233. Real zero: at 𝑥 = 2 (mult. of 1) and at

x=−1

3 (mult. of 1); Imaginary zeros: at 𝑥 =

𝑖 (mult. of 1), at 𝑥 = −𝑖 (mult. of 1)

234. 4 Real zeros: at 𝑥 = 1 (mult. of 1), at

𝑥 = 4 (mult. of 1), at 𝑥 = −2 (mult. of 1),

at 𝑥 = 2 (mult. of 1)

235. a. f(x) = (x + 2)(x – 7)(x – 1)

b. -2, 1, 7

c. -2, 1 , 7

d. Below, f(-3) is negative, or since

the degree is 3 and the leading

coefficient is positive.

e. Above, crosses at -2

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f. Below, crosses at 1

g. Above, crosses at 7

h. 236. 𝑓(𝑥) = (𝑥 + 3)(𝑥 + 2)(𝑥 − 2)

237. 𝑓(𝑥) = (𝑥 + 3)(𝑥 + 1)(𝑥 − 2)(𝑥 − 4)

238. 𝑓(𝑥) = (𝑥2 − 3) (𝑥 −1

3) (𝑥 + 5)

239. 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 3)(𝑥2 + 1) (𝑥 −3

5)

240. 𝑓(𝑥) = 𝑥(𝑥 − 2)2 241. 𝑓(𝑥) = (𝑥 − 1)2(𝑥 + 1)2

242. 𝑓(𝑥) = (𝑥 − 1) (𝑥 −3

4) (𝑥 − 2)

243. 𝑓(𝑥) = 𝑥(𝑥 + 1)(𝑥 − 3) 244. 𝑓(𝑥) = 𝑥2(𝑥 + 5)(𝑥 − 1) 245. 𝑓(𝑥) = (𝑥2 + 4)(𝑥 + 5)3 246. 𝑓(𝑥) = 𝑥(𝑥 − 1.5)(𝑥 + 1.5) 247. 𝑓(𝑥) = 𝑥(𝑥2 − 1)(𝑥2 − 4)

REVIEW

1. D

2. B

3. C

4. A

5. D

6. B

7. D

8. D

9. B

10. A

11. A

12. A

1. x = −2 (mult. of 2)

x = −1 (mult. of 1)

x = 0 (mult. of 1)

x = 1 (mult. of 1)

x = 3 (mult. of 1)

2. 3(x − 1)(x2 + 2x + 2)

3. x = −2 (mult. of 1)

x = 5 (mult. of 1)

x = 7 (mult. of 1)

4. f(x) = x(x2 − 1)(x2 − 20.25)

5. f(x) = (x + 4)(x + 2)(x – 1)(x – 3)(x – 5)