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. (−4𝑔3 ℎ2 𝑗 −2 )−3
4. (5r 3 s4 t 2 )(2r 3 s−3 )4
4𝑘 3
2
2. (3𝑚𝑛2 )
5. (3u2 v −4 )3 (6u4 v3 )−2
−2
3p7 q3
3. ((2p2 q2 )3 )
6. (
−3
8w2 x−3 y4 z5
)
12w3 x−4 y5 z−6
Properties of Exponents: Homework
Simplify the following expressions.
7. (−3𝑔−4 ℎ3 𝑗 −3 )−4
4𝑘 4
10. 4(5r10 s12 t 8 )(2r 4 s−5 )−3
2
8. (6𝑚3 𝑛−4 )
8p7 q9
11. (6u6 v −3 )3 (9u5 v−6 )−2
−2
6w−3 x−4 y5 z6
9. ((2p2 q2 )4 )
Algebra II - Polynomials
−2
12. (15w3 x−4 y5 z−6 )
~1~
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. 2𝑥 2 + 3𝑥 2
4
14. 7 𝑦 − 3𝑦 2 + 3𝑦
15. 5𝑎3 − 2𝑎 − 4𝑎 + 3
16.
6𝑎 2
𝑏
− 5𝑎𝑏 2 + 2𝑎𝑏 2
17. (2𝑥 −2 − 4) + (−5𝑥 −2 − 3)
Perform the indicated operations.
18. (4g 2 − 2) − (3g + 5) + (2g 2 − g)
19. (6𝑡 − 3𝑡 2 + 4) − (𝑡 2 + 5𝑡 − 9)
20. (7𝑥 5 + 8𝑥 4 − 3𝑥) + (5𝑥 4 + 2𝑥 3 + 9𝑥 − 1)
21. (−10𝑥 3 + 4𝑥 2 − 5𝑥 + 9) − (2𝑥 3 − 2𝑥 2 + 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(4𝑎2 𝑏 − 3𝑎𝑏 2 − 6𝑎𝑏)
27. (m − 3)(2m2 + 4m − 5)
24. 7𝑗𝑘 2 (5𝑗 3 𝑘 + 9𝑗 2 − 2𝑘 + 10)
28. (2𝑓 + 5)(6𝑓 2 − 4𝑓 + 1)
25. (2x − 3)(4x + 2)
29. (3t 2 − 2t + 9)(4t 2 − t + 1)
26. (𝑐 2 − 3)(𝑐 + 4)
Algebra II - Polynomials
~2~
NJCTL.org
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 = 𝜋𝑟 2 ℎ).
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
36. (2𝑑 + 4𝑒)2
35. (𝑐 − 1)(𝑐 − 1)
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
Algebra II - Polynomials
~3~
NJCTL.org
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. √2𝑥 2 + 0.4𝑥 3
4
44. 7𝑦 − 8𝑦 2 + 9𝑦
45. 11𝑎4 − 2𝑎3 + 7𝑎2 − 8𝑎 + 9
46.
6𝑎 2
11
−
5𝑎
9
+2
2
47. (2𝑥 3 − 4) + (−5𝑥 2 − 3)
Perform the indicated operations:
48. (3n − 13) − (2n2 + 4n − 6) − (5𝑛 − 4)
49. (5g 2 − 4) − (3g 3 + 7) + (5g 2 − 5g)
50. (−8𝑥 4 + 7𝑥 3 − 3𝑥 + 5) + (5𝑥 4 + 2𝑥 2 − 16𝑥 − 21)
51. (17𝑥 3 − 9𝑥 2 + 5𝑥 − 18) − (11𝑥 3 − 2𝑥 2 − 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(3𝑥 2 − 5𝑥 − 2)
58. (2𝑐 2 − 4)(3𝑐 + 2)
54. −6𝑎(3𝑎2 𝑏 − 5𝑎𝑏 2 − 7𝑏)
59. (2m − 5)(3m2 − 6m − 4)
55. 8𝑗 2 𝑘 3 (2𝑗 3 𝑘 + 6𝑗 2 − 5𝑘 + 11)
60. (3𝑓 + 4)(6𝑓 2 − 4𝑓 + 1)
56. (4x + 5)(6x + 1)
61. (2𝑝2 − 5)(𝑝2 + 8𝑝 + 2)
57. (2𝑏 − 9)(4𝑏 − 2)
62. (5t 2 − 3t + 6)(3t 2 − 2t + 1)
Algebra II - Polynomials
~4~
NJCTL.org
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
1
cylinder (V = 𝜋𝑟 2 ℎ).
3
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)
70. (3𝑑 − 5𝑒)2
68. (𝑏 − 2)2
71. (5𝑓 + 9)(5𝑓 + 9)
69. (𝑐 − 1)(𝑐 + 1)
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
Algebra II - Polynomials
~5~
NJCTL.org
Factoring I Classwork
Factoring I Homework
Factoring out the GCF
Factoring out the GCF
77. 6x3y2 – 3x2y
91. 8x3y – 4x2y2
78. 10p3q – 15p3q2 – 5p2q2
92. 8m3n3 – 4m2n3 – 32mn3
79. 7m3n3 – 7m3n2 + 14m3
93. -18p3q2 + 3pq
Factoring ax2 + bx + c
Factoring ax2 + bx + c
80. x2 – 5x – 24
94. m2 – 2m – 24
81. m2 – mn – 6n2
95. a2 – 13a + 12
82. x2 – 2xy + y2
96. n2 + n – 6
83. a2 + ab – 12b2
97. x2 – 10xy + 21y2
84. x2 – 6xy + 8y2
98. x2 + 11xy + 18y2
85. 2x2 + 7x + 3
99. 6x2 – 5x + 1
86. 6x2 – x – 2
100. 15p2 – 22p – 5
87. 5a2 + 17a – 12
101. 10m2 + 13m – 3
88. 6m2 - 5mn + n2
102. 12x2 – 7xy + y2
89. 6p2 + 37p + 6
103. 4p2 + 24p + 35
104. 15m2 – 13mn + 2n2
90. 4c2 + 20cd + 25d2
Spiral Review
105. Simplify:
106. Multiply:
5 – 4 [(-2) – (-2)]
Algebra II - Polynomials
2
3
4
∙4
2
3
107. Divide
2
3
4
÷4
2
3
~6~
108. Evaluate, use x = 5:
-2(-6x – 9) + 4
NJCTL.org
125. x4y + 12x3y + 20x2y
Factoring II Classwork
Factoring a2 – b2, a3 – b3, a3 + b3
Factoring II Homework
109. a3 – 1
Factoring a2 – b2, a3 – b3, a3 + b3
110. 25x2 – 16y2
126. y3 + 27
111. 121a2 – 16b2
127. 64m3 – 1
112. 27x3 + 8y3
128. p2 – 36q2
113. a3b3 – c3
129. m2n2 – 4
114. 4x2y2 – 1
130. x2 + 16
Factoring by Grouping
131. 8x3 – 27y3
115. 2xy + 5x + 8y + 20
Factoring by Grouping
116. 9mn – 3m – 15n + 5
132. 6mp – 2m – 15p + 5
117. 2xy – 10x – 3y + 15
133. 6xy + 15x + 4y + 10
118. 10rs – 25r + 6s – 15
134. 4rs – 4r + 3s – 3
119. 10pq – 2p – 5q + 1
135. 6tr – 9t – 2r + 3
120. 10mn + 5m + 6n + 3
136. 8mn + 4m + 6n + 3
137. 3xy – 4x – 15y + 20
Mixed Factoring
121. 3x3 – 12x2 + 36x
Mixed Factoring
122.
6m3
+
4m2
– 2m
138. 3m3 – 3mn2
123. 3a3b – 48ab
139. -6x3 – 28x2 + 10x
124. 54x4 + 2xy3
Algebra II - Polynomials
~7~
NJCTL.org
140. 18a3b – 50ab
143. 2x2y2 – 2x2y – 2xy2 + 2xy
141. x4y + 27xy
142. -12r3 – 21r2 – 9r
Spiral Review
144. Simplify:
8(-4)  (2)(-1) + (4)2
145. Simplify:
146. Add:
172 - (12 - 4)2 + 2
2
3
7
5
2 +5
147. Evaluate, use x = -3, y = 2
-3x + 2y – xy + x
Division of Polynomials: Class Work
Simplify.
148.
6x3 −3x2 +9x
3x
149. (4𝑎4 𝑏 3 + 8𝑎3 𝑏 3 − 6𝑎2 𝑏 2 ) ÷ (2𝑎2 𝑏)
150.
6x3 −4x2 +7x+3
3x+1
151. (4𝑎4 + 8𝑎3 − 6𝑎2 + 3𝑎 + 4) ÷ (𝑎 − 1)
Algebra II - Polynomials
~8~
NJCTL.org
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) = 𝑥 3 + 𝑥 2 − 10𝑥 − 10.
Is 𝑥 + 1 one of the factors of 𝑃? Explain.
155. The volume a hexagonal prism is (3𝑡 3 − 4𝑡 2 + 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. (4𝑎 4 𝑏3 + 8𝑎 3 𝑏 3 − 16𝑎 2 𝑏 2 ) ÷ (4𝑎𝑏 2 )
Algebra II - Polynomials
~9~
NJCTL.org
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) = 2𝑥 3 + 5𝑥 2 − 12𝑥 + 5.
Is 𝑥 − 1 one of the factors of 𝑃? Explain.
Algebra II - Polynomials
~10~
NJCTL.org
163. (8f 3 )(2f + 4)−1
164. The volume a hexagonal prism is (4𝑡 3 − 3𝑡 2 + 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) = 𝑥 4 + 3𝑥 3 − 28𝑥 2 − 36𝑥 + 144.
a. Is 1 a zero of the polynomial 𝑃?
b. Is 𝑥 + 3 one of the factors of 𝑃?
Algebra II - Polynomials
~11~
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) = 2𝑥 4 + 3𝑥 2 − 2
171.
𝑦 = 5𝑥 5 − 3𝑥 + 1
172.
𝑔(𝑥) = −2𝑥(4𝑥 2 − 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.
Algebra II - Polynomials
176.
177.
~12~
178.
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) = 5𝑥 4 − 6𝑥 2 + 3𝑥
184. 𝑦 = 5𝑥 5 − 3𝑥 3 + 1𝑥
185. 𝑔(𝑥) = 2𝑥 2 (4𝑥 3 − 3𝑥)
4
186. ℎ(𝑥) = − 5 𝑥 2 + 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.
Algebra II - Polynomials
~13~
NJCTL.org
194.
196.
195.
x
-2
-1
0
1
2
3
4
f(x)
5
1
-1
0
2
1
-1
x
-2
-1
0
1
2
3
4
197.
f(x)
-4
0
2
1
-1
-3
-1
198.
x
-2
-1
0
1
2
3
4
f(x)
2
-3
-4
-1
2
5
-2
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.
x
-2
-1
0
1
2
3
4
f(x)
6
2
1
3
1
-1
0
Algebra II - Polynomials
204.
205.
x
-2
-1
0
1
2
3
4
f(x)
2
4
2
-2
0
3
1
~14~
x
-2
-1
0
1
2
3
4
f(x)
4
-2
-3
-1
1
3
7
NJCTL.org
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)
212. ℎ(𝑥) = 𝑥 2 (𝑥 − 10)(𝑥 + 1)
210. 𝑔(𝑥) = (𝑥 2 − 1)(𝑥 2 + 1)
213. 𝑦 = (𝑥 2 − 9)(𝑥 + 3)2 (𝑥 2 + 9)
211. 𝑦 = (𝑥 + 1)2 (𝑥 + 2)(𝑥 − 2)
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.
3rd degree
Algebra II - Polynomials
216.
4th degree
~15~
6th degree
NJCTL.org
Name all of the real and imaginary zeros and state their multiplicity.
217. 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 3)(𝑥 + 3)(𝑥 − 3)
220. ℎ(𝑥) = 𝑥 3 (𝑥 − 7)(𝑥 − 6)𝑥(2𝑥 + 4)(𝑥 − 5)
218. 𝑔(𝑥) = (𝑥 2 − 4)(𝑥 2 + 4)
221. 𝑦 = (𝑥 + 4)2 (𝑥 2 − 16)(𝑥 2 + 16)
219. 𝑦 = (𝑥 + 7)2 (4𝑥 2 − 64)
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) = 2𝑥 3 + 16𝑥 2 + 30𝑥
225. 𝑓(𝑥) = 𝑥 4 − 8𝑥 2 − 9
223. 𝑓(𝑥) = 𝑥 4 + 9𝑥 2
226. 𝑓(𝑥) = 2𝑥 3 + 𝑥 2 − 16𝑥 − 15
224. 𝑓(𝑥) = 2𝑥 3 + 3𝑥 2 − 8𝑥 − 12
227. 𝑓(𝑥) = 𝑥 3 + 4𝑥 2 − 25𝑥 − 100
228. Consider the function 𝑓(𝑥) = 𝑥 3 + 3𝑥 2 − 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?
Algebra II - Polynomials
~16~
NJCTL.org
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) = 𝑥 3 − 3𝑥 2 − 2𝑥 + 6
232. 𝑓(𝑥) = 𝑥 4 − 𝑥 2 − 30
230. 𝑓(𝑥) = 𝑥 4 + 𝑥 2 − 12
233. 𝑓(𝑥) = 3𝑥 4 − 5𝑥 3 + 𝑥 2 − 5𝑥 − 2
231. 𝑓(𝑥) = 𝑥 3 + 5𝑥 2 − 9𝑥 − 45
234. 𝑓(𝑥) = 𝑥 4 − 5𝑥 3 + 20𝑥 − 16
235. Consider the function 𝑓(𝑥) = 𝑥 3 − 6𝑥 2 − 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.
Algebra II - Polynomials
~17~
NJCTL.org
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
Algebra II - Polynomials
~18~
NJCTL.org
238. ±√3,
1
,
3
−5
241.
3
239. 2, 3, 𝑖, −𝑖, 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.
Algebra II - Polynomials
~19~
NJCTL.org
UNIT REVIEW
Multiple Choice
6p8 q9
−2
1. Simplify the following expression: ((2p3 q4 )3 )
a.
b.
c.
d.
3
4pq3
9
16p2 q6
4pq3
3
16p2 q6
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 + 4x 2 )ft 2
b. (30x + x 2 )ft 2
c. (200 + 60x + 4x 2 )ft 2
d. (200 + 30x + x 2 )ft 2
5. What is the area of a square with sides (6x – 2) inches?
a. (36x 2 − 4) in2
b. (36x 2 + 4) in2
c. (36x 2 − 12x − 4) in2
d. (36x 2 − 24x + 4) in2
6.
7.
27w3 x5 −12w4 x3 +24w3 x2
is equivalent
6w2 x2
9wx3 −4w2 x+4w
a.
3
9wx3
b. 2 − 2w 2 x + 4w
9wx3 −4w2 x
c.
+ 4w
3
3
2
9wx +4w x+8w
d.
2
2
(2𝑎4
(𝑎
to which of the following?
− 6𝑎 + 4) ÷ − 2)
a. 2𝑎3 − 3𝑎 − 2
b. 2𝑎3 − 3𝑎2 − 2
c. 2𝑎3 + 4𝑎2 − 2𝑎 − 4 +
d. 2𝑎3 + 4𝑎2 + 2𝑎 + 4 +
Algebra II - Polynomials
−4
𝑎−2
12
𝑎−2
~20~
NJCTL.org
8. A box has volume of (3x 2 − 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 = 3𝑥 5 − 2𝑥
b. 𝑦 = 5𝑥 7 − 3𝑥 3 + 9
c. 𝑦 = 𝑥 5 (𝑥 7 + 𝑥 5 )
d. 𝑦 = 7𝑥 10
x
f(x)
11. What value should A be in the table so that the function has 4 zeros?
-2
6
a. -2
-1
A
b. 0
0
2
c. 1
1
3
d. 3
2
1
3
-1
12. Name all of the real and imaginary zeros and state their multiplicity:
4
0
𝑦 = (𝑥 2 + 8𝑥 + 16)(4𝑥 2 + 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.
Algebra II - Polynomials
~21~
NJCTL.org
2. Given the function 𝑓(𝑥) = 3𝑥 3 + 3𝑥 2 − 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) = 𝑥 3 − 10𝑥 2 + 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.
Algebra II - Polynomials
~22~
NJCTL.org
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Answer Key
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
𝑗6
−64𝑔9 ℎ6
16𝑘 6
9𝑚2 𝑛4
64𝑞 6
9𝑝2
80𝑟 15 𝑡 2
𝑠8
3
4𝑢2 𝑣 18
27𝑤 3 𝑦 3
8𝑥 3 𝑧 33
𝑔16 𝑗 12
81ℎ12
4𝑘 8 𝑛8
9𝑚6
4𝑝2
𝑞2
5𝑠 27 𝑡 8
2𝑟 2
8𝑢8 𝑣 3
39. 32x5+320x4y+1280x3y2+2560x2y3+2560xy4+1024y5
3
25𝑤 12
41. 729x6-5832x5z+19440x4z2-34560x3z3+34560x2z4-
40.
343a3+147a2b+21ab2+b3
18432xz5+4096z6
4𝑧 24
Yes, 5x2, degree: 2, monomial/quadratic,
5
4
Yes, -3y 2 +3 y, degree: 2,
7
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. -4t 2 +t+13
20. 7𝑥 5 + 13𝑥 4 + 2𝑥 3 + 6𝑥 −1
21. −12𝑥 3 + 6𝑥 2 − 6𝑥 − 3
22. Perimeter = (6x2+12x-1) inches
23. -8a3b+6a2b2+12a2b
24. 35j4k3+63j3k2-14jk3+70jk2
Algebra II - Polynomials
~23~
42. y4-20y3z+150y2z2-500yz3+625z4
Yes, 0.4𝑥 3 + √2𝑥 2 , 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
43.
NJCTL.org
71.
16j5k4+48j4k3-40j2k4+88j2k3
24x2+34x+5
8b2-40b+18
6c3+4c2-12c-8
6m3-27m2+22m+20
18f3+12f2-13f+4
2p4+16p3-p2-40p-10
15t4-19t3+29t2-15t+6
Area = (12x2-29x+15) in.2
Area = 81x3-27x+6) m2
Area = (3x2+33x+8) in.2
Areadeck = (4x2+146x+276) ft.2
9a2-1
b2-4b+4
c2-1
9d2-30de+25e2
25f2+90f+81
72.
Area = (16x2-48xy+36y2) in.2
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
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.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
73. 64a6-192a5b+240a4b2-160a3b3+60a2b4-12ab5+b6
74.
112.
27x3+54x2y+36xy2+8y3
113.
75. 3125y5-12500y4z+20000y3z2-16000y2z3+6400yz4-1024z5
114.
76. a4+28a3b+294a2b2+1372ab3+2401b4
115.
77. 3x2y(2xy – 1)
116.
78. 5p2q(2p – 3pq – q)
117.
79. 7m3(n3 – n2 + 2)
118.
80. (x – 8)(x + 3)
119.
81. (m – 3n)(m + 2n)
120.
82. (x – y)(x – y)
121.
83. (a + 4b)(a – 3b)
122.
84. (x – 4y)(x – 2y)
123.
85. (2x + 1)(x + 3)
124.
86. (3x – 2)(2x + 1)
125.
87. (5a – 3)(a + 4)
126.
88. (2m – n)(3m – n)
127.
89. (6p + 1)(p + 6)
128.
90. (2c + 5d)(2c + 5d)
129.
91. 4x2y(2x-y)
130.
92. 4mn3(2m2-m-8)
131.
93. 3pq(-6p2q+1)
Algebra II - Polynomials
132.
~24~
(3p – 5)(5p + 1)
(2m + 3)(5m – 1)
(3x – y)(4x – y)
(2p + 7)(2p + 5)
(3m – 2n)(5m – n)
5
77
6
33
56
82
(a – 1)(a2 + a + 1)
(5x – 4y)(5x + 4y)
(11a – 4b)(11a + 4b)
(3x + 2y)(9x2 + 6xy + 4y2)
(ab – c)(a2b2 + abc + c2)
(2xy – 1)(2xy + 1)
(x + 4)(2y + 5)
(3m – 5)(3n – 1)
(2x – 3)(y – 5)
(5r + 3)(2s – 5)
(2p – 1)(5q – 1)
(5m + 3)(2n + 1)
3x(x – 6)(x + 2)
2m(3m – 1)(m + 1)
3ab(a – 4)(a + 4)
2x(3x + y)(9x2 – 3xy + y2)
x2y(x + 10)(x + 2)
(y + 3)(y2 – 3y + 9)
(4m – 1)(16m2 + 4m + 1)
(p – 6q)(p + 6q)
(mn – 2)(mn + 2)
Not Factorable
(2x – 3y)(4x2 + 6xy + 9y2)
(2m – 5)(3p – 1)
NJCTL.org
133.
134.
135.
136.
137.
138.
139.
140.
141.
142.
143.
(3x + 2)(2y + 5)
(4r + 3)(s – 1)
(3t – 1)(2r – 3)
(4m + 3)(2n + 1)
(x – 5)(3y – 4)
3m(m – n)(m + n)
-2x(3x – 1)(x + 5)
2ab(3a – 5)(3a + 5)
xy(x + 3)(x2 – 3x + 9)
-3r(4r + 3)(r + 1)
2xy(x – 1)(y – 1)
145.
32
227
146.
7
147.
150.
16
2x2-x+3
2a2b2+4ab2-3b
2x2 – 2x + 3
151.
4a3+12a2+6a+9 +
152.
a. 3𝑥 + 14 +
153.
a. 𝑥 2 − 4𝑥 + 10 −
154.
Yes, because P(-1) = 0.
155.
B = (3t2-7t + 8 -
156.
4x3-3x+6
a3b+2a2b- 4a
144.
148.
149.
157.
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
165.
31
35
6
4
𝑓
𝑓2
13
𝑎−1
24
b. 24
𝑥−2
6
2
𝑥+1
b. -2
) cm.2
𝑡+1
158.
f+ -
159.
3x2-12x+45 -
160.
a. 𝑥 2 + 2𝑥 + 4 −
161.
a. 𝑥 2 + 4𝑥 − 12 +
162.
Yes, because P(1) = 0.
163.
4f2-8f+16 -
164.
height = 4t2+t+3+
133
𝑥+3
16
𝑥−2
19
b. -16
𝑥+1
b. 19
32
𝑓+2
Algebra II - Polynomials
5
𝑡−1
cm
~25~
NJCTL.org
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
Algebra II - Polynomials
~26~
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)
205.
NJCTL.org
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
5
x=- (mult. of 1), at x=3 (mult. of 1)
2
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
218.
Algebra II - Polynomials
~27~
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)
2 Real zeros: at 𝑥 = √6 (mult. of 1), at
𝑥 = −√6 (mult. of 1);
232.
2 Imaginary zeros: at 𝑥 = 𝑖√5 (mult. of
233.
1), at 𝑥 = −𝑖√5 (mult. of 1)
Real zero: at 𝑥 = 2 (mult. of 1) and at
1
x=− (mult. of 1); Imaginary zeros: at 𝑥 =
3
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
NJCTL.org
f. Below, crosses at 1
g. Above, crosses at 7
h.
236.
237.
𝑓(𝑥) = (𝑥 + 3)(𝑥 + 1)(𝑥 − 2)(𝑥 − 4)
238.
𝑓(𝑥) = (𝑥 2 − 3) (𝑥 − ) (𝑥 + 5)
239.
𝑓(𝑥) = (𝑥 − 2)(𝑥 − 3)(𝑥 2 + 1) (𝑥 − 5)
240.
241.
𝑓(𝑥) = 𝑥(𝑥 − 2)2
𝑓(𝑥) = (𝑥 − 1)2 (𝑥 + 1)2
242.
𝑓(𝑥) = (𝑥 − 1) (𝑥 − 4) (𝑥 − 2)
243.
244.
245.
246.
247.
𝑓(𝑥) = 𝑥(𝑥 + 1)(𝑥 − 3)
𝑓(𝑥) = 𝑥 2 (𝑥 + 5)(𝑥 − 1)
𝑓(𝑥) = (𝑥 2 + 4)(𝑥 + 5)3
𝑓(𝑥) = 𝑥(𝑥 − 1.5)(𝑥 + 1.5)
𝑓(𝑥) = 𝑥(𝑥 2 − 1)(𝑥 2 − 4)
1
3
3
3
𝑓(𝑥) = (𝑥 + 3)(𝑥 + 2)(𝑥 − 2)
REVIEW
1. D
2. B
3. C
4. A
5.
6.
7.
8.
D
B
D
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)(x 2 + 2x + 2)
3. x = −2 (mult. of 1)
x = 5 (mult. of 1)
x = 7 (mult. of 1)
4. f(x) = x(x 2 − 1)(x 2 − 20.25)
5. f(x) = (x + 4)(x + 2)(x – 1)(x – 3)(x – 5)
Algebra II - Polynomials
~28~
NJCTL.org