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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
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 )
Algebra II - Polynomials ~2~ NJCTL.org
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
Algebra II - Polynomials ~3~ NJCTL.org
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 )
Algebra II - Polynomials ~4~ NJCTL.org
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
Algebra II - Polynomials ~5~ NJCTL.org
74. (3 x+2 y )3
75. (5 y−4 z )5
76. (a+7b )4
Algebra II - Polynomials ~6~ NJCTL.org
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:
Algebra II - Polynomials ~7~ NJCTL.org
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
Algebra II - Polynomials ~8~ NJCTL.org
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
Algebra II - Polynomials ~9~ NJCTL.org
151. (4a4+8a3−6a2+3a+4 ) ÷ (a−1 )
Algebra II - Polynomials ~10~ NJCTL.org
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 )
Algebra II - Polynomials ~11~ NJCTL.org
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.
Algebra II - Polynomials ~12~ NJCTL.org
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?
Algebra II - Polynomials ~13~ NJCTL.org
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.
Algebra II - Polynomials ~14~ 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 )=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.
Algebra II - Polynomials ~15~ NJCTL.org
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.
Algebra II - Polynomials ~16~ NJCTL.org
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.
Algebra II - Polynomials ~17~ NJCTL.org
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?
Algebra II - Polynomials ~18~ NJCTL.org
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
Algebra II - Polynomials ~19~ NJCTL.org
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.
Algebra II - Polynomials ~20~ NJCTL.org
237.−3 ,−1,2 ,4
Algebra II - Polynomials ~21~ NJCTL.org
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.
Algebra II - Polynomials ~22~ NJCTL.org
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
Algebra II - Polynomials ~23~ NJCTL.org
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.
Algebra II - Polynomials ~24~ NJCTL.org
x f(x)-2 6-1 A0 21 32 13 -14 0
Algebra II - Polynomials ~25~ NJCTL.org
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.
Algebra II - Polynomials ~26~ NJCTL.org
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
Algebra II - Polynomials ~27~ NJCTL.org
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)
Algebra II - Polynomials ~28~ NJCTL.org
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
Algebra II - Polynomials ~29~ NJCTL.org
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
Algebra II - Polynomials ~30~ NJCTL.org
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
Algebra II - Polynomials ~31~ NJCTL.org
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)
Algebra II - Polynomials ~32~ NJCTL.org