polynomial functionscontent.njctl.org/courses/math/algebra-ii/polynomial... · web view2014/10/14...

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 g3h2 j−2 )−3

2. ( 4k3

3mn2 )2

3. ( 3 p7q3

(2 p2q2 )3 )−2

4. (5 r3 s4 t 2) (2 r3 s−3 )4

5. (3u2 v−4 )3 (6u4 v3 )−2

6. ( 8w2 x−3 y4 z5

12w3 x−4 y5 z−6 )−3

Properties of Exponents: HomeworkSimplify the following expressions.

7. (−3 g−4h3 j−3 )−4

8. ( 4k 4

6m3n−4 )2

9. ( 8 p7q9

(2 p2q2 )4 )−2

10. 4 (5 r10 s12 t8 ) (2 r4 s−5 )−3

11. (6u6 v−3 )3 (9u5 v−6 )−2

Algebra II - Polynomials ~1~ NJCTL.org

https://www.engageny.org/ccss-library

12. ( 6w−3 x−4 y5 z6

15w3 x−4 y5 z−6 )−2

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. 2 x2+3 x2

14.47

y−3 y2+3 y

15. 5a3−2a−4a+3

16. 6a2

b−5ab2+2ab2

17. (2 x−2−4 )+(−5x−2−3 )

Perform the indicated operations. 18. (4 g2−2 )−(3g+5 )+(2 g2−g )

19. (6 t−3 t 2+4 )−(t2+5 t−9)

20. (7 x5+8 x4−3 x )+(5 x4+2x3+9 x−1)

21. (−10 x3+4 x2−5 x+9 )−(2 x3−2 x2+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 (4 a2b−3ab2−6 ab )

24. 7 jk 2 (5 j3k+9 j2−2k+10 )

25. (2 x−3 ) (4 x+2 )

26. (c2−3 ) (c+4 )

27. (m−3 ) (2m2+4m−5 )

28. (2 f +5 ) (6 f 2−4 f +1 )


29. (3 t 2−2 t+9 ) (4 t 2−t +1 )

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 = π r2h).

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+4 e )2

37. (5 f +9 ) (5 f −9 )

38. What is the area of a square with sides (3x+2) inches?

Expand, using the Binomial Theorem:

39. (2 x+4 y )5

40. (7a+b )3


41. (3 x−4 z )6

42. ( y−5 z )4

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+0.4 x3

44.47 y

−8 y2+9 y

45. 11a4−2a3+7 a2−8a+9

46. 6a2

11−5a9

+2

47. (2x23−4)+(−5 x2−3 )

Perform the indicated operations:

48. (3n−13 )−(2n2+4n−6 )−(5n−4)

49. (5 g2−4 )−(3 g3+7 )+ (5g2−5g )

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

51. (17 x3−9 x2+5x−18 )−(11 x3−2 x2−19x+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. 4 x (3 x2−5 x−2 )

54. −6a (3 a2b−5ab2−7b )

55. 8 j2k3 (2 j3k+6 j2−5 k+11)

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


57. (2b−9 ) (4b−2 )

58. (2c2−4 ) (3c+2 )

59. (2m−5 ) (3m2−6m−4 )

60. (3 f +4 ) (6 f 2−4 f +1 )

61. (2 p2−5 ) ( p2+8 p+2 )

62. (5 t 2−3 t+6 ) (3 t2−2 t+1 )

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 = 13

π r2h).

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. (5 f +9 ) (5 f +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. (3 x+2 y )3

75. (5 y−4 z )5

76. (a+7b )4


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)] 2 34∙ 4 23 2 34

÷ 4 23 -2(-6x – 9) + 4

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

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 + 2 227+5 35 -3x + 2y – xy + x

Division of Polynomials: Class Work Simplify.

148. 6 x3−3x2+9 x3 x

149. (4a4b3+8a3b3−6a2b2 ) ÷ (2a2b )

150. 6 x3−4 x2+7 x+33 x+1


151. (4a4+8a3−6a2+3a+4 ) ÷ (a−1 )


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

a. Divide f by x−2. b. Find f (2).

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

a. Divide g by x+1.

b. Find g(−1).

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

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

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

Division of Polynomials: HomeworkSimplify.

156. 16x5−12x3+24 x2

4 x2

157. (4a4b3+8a3b3−16a2b2 ) ÷ (4a b2 )


158. (3 f 3+18 f −12 ) (3 f 2 )−1

159. 3x3−3x2+9 x+2x+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+5 x2−12x+5.

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


163. (8 f 3 ) (2 f +4 )−1

164. The volume a hexagonal prism is (4 t3−3 t2+2 t+2 ) c m3 . 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+3 x3−28 x2−36 x+144.

a. Is 1 a zero of the polynomial P?

b. Is x+3 one of the factors of P?


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+3 x2−2

171. y=5 x5−3 x+1

172. g ( x )=−2 x (4 x2−3 x)

173. h ( x )=4 x

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.


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 )=5 x4−6 x2+3 x

184. y=5 x5−3 x3+1x

185. g ( x )=2 x2 (4 x3−3x )

186. h ( x )=−45

x2+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.

194. 195.

196. 197. 198.

Analyzing Graphs and Tables of Polynomial Functions: HomeworkIdentify 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 10 -11 02 23 14 -1

x f(x)-2 2-1 -30 -41 -12 23 54 -2

x f(x)-2 -4-1 00 21 12 -13 -34 -1

x f(x)-2 2-1 40 21 -22 03 34 1

x f(x)-2 6-1 20 11 32 13 -14 0

x f(x)-2 4-1 -20 -31 -12 13 34 7

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

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(4 x2−64)

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+16 x2+30 x 225. f ( x )=x4−8 x2−9

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

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

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

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 )=x3−3x2−2 x+6 232. f ( x )=x4−x2−30

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

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


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

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


238. ±√3 , 13

,−5 241.

239. 2 ,3 ,i ,−i , 35

Writing Polynomials from Given Zeros: Homework Write a polynomial function of least degree with integral coefficients that has the given zeros.

242. 1 ,2 , 34 246.

243. −1 ,3 ,0

244. 0 (mult .2 ) ,−5 ,1

245. −2 i ,2 i ,−5(mult .3) 247.


UNIT REVIEWMultiple Choice

1. Simplify the following expression: ( 6 p8q9

(2 p3q4 )3 )−2

a.3

4 p q3

b.9

16 p2q6

c. 4 p q3

3

d. 16 p2q6

92. The sides of a rectangle are (2x2 – 11x +1) ft and (3x – 4) ft find the perimeter of the rectangle.a. (2x2 – 8x – 3) ftb. (4x2 – 16x – 6) c. (5x3 – 11x – 3) ftd. (6x3 – 41x2 + 47x – 4) ft23. 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) ft24. 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. (60 x+4 x2 ) ft2

b. (30 x+x2 ) ft2

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

d. (200+30 x+x2 ) ft25. What is the area of a square with sides (6x – 2) inches?a. (36 x2−4 )¿2 b. (36 x2+4 )¿2

c. (36 x2−12 x−4 )¿2

d. (36 x2−24 x+4 )¿2

6. 27w3 x5−12w4 x3+24w3 x2

6w2x2 is equivalent to which of the following?

a. 9w x3−4w2 x+4w3

b. 9w x3

2−2w2 x+4w

c. 9w x3−4w2 x3

+4w


d. 9w x3+4w2x+8w27. (2a4−6a2+4 )÷ (a−2 )

a. 2a3−3a−2b. 2a3−3a2−2

c. 2a3+4a2−2a−4+ −4a−2

d. 2a3+4a2+2a+4+ 12a−2

8. A box has volume of (3 x2−2 x−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; positiveb. odd degree; negativec. even degree; positived. even degree; negative10.Which of the following equations is of an odd-function?a. y=3 x5−2 xb. y=5 x7−3 x3+9c. y=x5 ( x7+ x5 )d. y=7 x1011.What value should A be in the table so that the function has 4 zeros?a. -2b. 0c. 1d. 3

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

y=( x2+8 x+16 )(4 x2+64)a. Real zeros: -4 with multiplicity 2; Imaginary zeros: ± 4i each with multiplicity 1b. Real zeros: -4 with multiplicity 3, 4 with multiplicity 1; No imaginary zerosc. Real zeros: -4 with multiplicity 4; No imaginary zerosd. Real zeros: -4 with multiplicity 2; Imaginary zeros: 2i with multiplicity 2

Extended Response1. Graph y=¿Name the real zeros and their multiplicity.


x f(x)-2 6-1 A0 21 32 13 -14 0

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

3. Name all of the real and imaginary zeros and state their multiplicity of the functionf ( x )=x3−10x2+11 x+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.


Answer Key

1. j6

−64 g9h6

2. 16k6

9m2n4

3. 64q6

9 p2

4. 80 r15 t 2

s8

5. 3

4u2v18

6. 27w3 y3

8 x3 z33

7. g16 j12

81h12

8. 4 k8n8

9m6

9. 4 p2

q2

10. 5 s27t 8

2 r2

11. 8u8 v3

3

12. 25w12

4 z24

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

14. Yes, -3 y2+347 y , degree: 2,

binomial/quadratic, -315. Yes, 5a3-6a+3, degree: 3, trinomial/cubic,

516. Not a polynomial function17. Not a polynomial function18. 6g2-4g-719. -4 t 2+t+1320. 7 x5+13x4+2x3+6 x−¿121. −12 x3+6 x2−6 x−3

22. Perimeter = (6x2+12x-1) inches23. -8a3b+6a2b2+12a2b24. 35j4k3+63j3k2-14jk3+70jk2

25. 8x2-8x-626. c3+4c2-3c-1227. 2m3-2m2-17m+1528. 12f3+22f2-18f+529. 12t4-11t3+41t2-11t+930. 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+435. c2-2c+136. 4d2+16de+16e2

37. 25f2-8138. (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, 0.4 x3+√2x2 , degree: 3, binomial/cubic, 0.4

44. Not a polynomial function45. Yes, already in std form, degree: 4, no

specific name/quartic, 1146. Yes, already in std form, degree: 2,

trinomial/quadratic, 6/1147. Not a polynomial function48. -2n2-6n-349. -3g3+10g2-5g-1150. -3x4+7x3+2x2-19x-1651. 6x3-7x2+24x-33


52. Perimeter = (10x2+24x-10) inches53. 12x3-20x2-8x54. -18a3b+30a2b2+42ab55. 16j5k4+48j4k3-40j2k4+88j2k3

56. 24x2+34x+557. 8b2-40b+1858. 6c3+4c2-12c-859. 6m3-27m2+22m+2060. 18f3+12f2-13f+461. 2p4+16p3-p2-40p-1062. 15t4-19t3+29t2-15t+663. 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. 9a2-168. b2-4b+469. c2-170. 9d2-30de+25e2

71. 25f2+90f+8172. 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

106.776

107.3356

108. 82109. (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 Factorable131. (2x – 3y)(4x2 + 6xy + 9y2)132. (2m – 5)(3p – 1)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. 32145. 227

146. 73135147. 16148. 2x2-x+3149. 2a2b2+4ab2-3b150. 2x2 – 2x + 3

151. 4a3+12a2+6a+9 + 13a−1

152. a. 3 x+14+ 24x−2 b. 24

153. a. x2−4 x+10− 2x+1 b. -2

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

155. B = (3t2-7t + 8 - 6t+1 ) cm.2

156. 4x3-3x+6157. a3b+2a2b- 4a

158. f+6f - 4f 2

159. 3x2-12x+45 - 133x+3

160. a. x2+2 x+4− 16x−2 b. -16

161. a. x2+4 x−12+ 19x+1 b. 19

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

163. 4f2-8f+16 - 32f +2

164. height = 4t2+t+3+ 5t−1 cm

165. a. No b. Yes166. Odd; positive; neither167. Even; negative; even168. Even; positive; neither169. Odd; negative; neither170. Even function171. Neither 172. Neither173. Odd174. 170: ∞ , ∞ 171: ∞ ,−∞ 172: −∞ ,∞

173: ∞ ,−∞

175. Odd function; 3 zeros176. Even function; 2 zeros177. Neither; 3 zeros178. Even function; 2 zeros179. Odd; negative; neither180. Even; negative; even181. Even; positive; even182. Odd; negative; odd183. Neither184. Odd function185. Odd function186. Even function187. 184: ∞ , ∞ 185: ∞ ,−∞ 186: ∞ ,−∞

187: −∞ ,−∞

188. Even function; 2 zeros189. Odd function; 1 zero190. Neither; 2 zeros


191. Odd function; 1 zero192. 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 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 min200. 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)

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=-52 (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, 1c. -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 -3f. Below, crosses at -1g. 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=2 i (mult. of 1), at x=−2 i (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=−13 (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 , 7d. Below, f(-3) is negative, or since

the degree is 3 and the leading coefficient is positive.

e. Above, crosses at -2f. Below, crosses at 1g. 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−13 ) ( x+5 )

239. f ( x )= (x−2 )(x−3)(x2+1)(x−35 )

240. f ( x )=x ¿241. f ( x )=(x−1)2¿

242. f ( x )= (x−1 )(x−34 ) (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)

REVIEW1. D2. B3. C4. A

5. D6. B7. D8. D

9. B10.A11.A12.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+2 x+2)3. x=−2 (mult . of 1 )

x=5 (mult .of 1 )x=7 (mult . of 1 )

4. f ( x )=x (x¿¿2−1)(x2−20.25)¿

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


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