Polynomial Functions

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. ( 4g 3 h 2 j 2 ) 3 4. (5r 3 s 4 t 2 )(2r 3 s 3 ) 4 2. ( 4k3 3mn 2)2 5. (3u 2 v 4 ) 3 (6u 4 v 3 ) 2 3. ( 3p7 q 3 (2p 2 q 2 ) 3) 2 6. ( 8w2 x 3 y 4 z 5 12w 3 x 4 y 5 z 6) 3 Properties of Exponents: Homework Simplify the following expressions. 7. ( 3g 4 h 3 j 3 ) 4 10. 4(5r 10 s 12 t 8 )(2r 4 s 5 ) 3 8. ( 4k4 6m 3 n 4)2 11. (6u 6 v 3 ) 3 (9u 5 v 6 ) 2 9. ( 8p7 q 9 (2p 2 q 2 ) 4) 2 12. ( 6w 3 x 4 y 5 z 6 15w 3 x 4 y 5 z 6) 2 Algebra II - Polynomials ~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 14. 4 7 y 3y2 + 3y 15. 5a 3 2a 4a + 3 16. 6a2 b 5ab2 + 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 (3x 2 + 4x +2) inches and the base is (4x-5) inches. Find the perimeter of the triangle. 23. 2a(4a 2 b 3ab 2 6ab) 27. (m 3)(2m 2 + 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 44. 4 7y 8y2 + 9y 45. 11a 4 2a 3 + 7a 2 8a + 9 46. 6a2 11 5a 9 + 2 47. (2x 2 3 4) + ( 5x 2 3) Perform the indicated operations: 48. (3n 13) (2n 2 + 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 (5x 2 +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(3a 2 b 5ab 2 7b) 59. (2m 5)(3m 2 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. (2p 2 5)(p 2 + 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 cylinder (V = 1 3 πr2 h). 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 Algebra II - Polynomials ~5~ NJCTL.org

Factoring I Classwork Factoring out the GCF 77. 6x 3 y 2 3x 2 y Factoring I Homework Factoring out the GCF 91. 8x 3 y 4x 2 y 2 78. 10p 3 q 15p 3 q 2 5p 2 q 2 92. 8m 3 n 3 4m 2 n 3 32mn 3 79. 7m 3 n 3 7m 3 n 2 + 14m 3 93. -18p 3 q 2 + 3pq Factoring ax 2 + bx + c 80. x 2 5x 24 Factoring ax 2 + bx + c 94. m 2 2m 24 81. m 2 mn 6n 2 95. a 2 13a + 12 82. x 2 2xy + y 2 96. n 2 + n 6 83. a 2 + ab 12b 2 97. x 2 10xy + 21y 2 84. x 2 6xy + 8y 2 98. x 2 + 11xy + 18y 2 85. 2x 2 + 7x + 3 99. 6x 2 5x + 1 86. 6x 2 x 2 100. 15p 2 22p 5 87. 5a 2 + 17a 12 101. 10m 2 + 13m 3 88. 6m 2-5mn + n 2 102. 12x 2 7xy + y 2 89. 6p 2 + 37p + 6 103. 4p 2 + 24p + 35 90. 4c 2 + 20cd + 25d 2 104. 15m 2 13mn + 2n 2 Spiral Review 105. Simplify: 106. Multiply: 107. Divide 108. Evaluate, use x = 5: 5 4 [(-2) (-2)] 2 3 4 4 2 3 2 3 4 4 2 3-2(-6x 9) + 4 Algebra II - Polynomials ~6~ NJCTL.org

Factoring II Classwork Factoring a 2 b 2, a 3 b 3, a 3 + b 3 109. a 3 1 110. 25x 2 16y 2 111. 121a 2 16b 2 112. 27x 3 + 8y 3 125. x 4 y + 12x 3 y + 20x 2 y Factoring II Homework Factoring a 2 b 2, a 3 b 3, a 3 + b 3 126. y 3 + 27 127. 64m 3 1 113. a 3 b 3 c 3 114. 4x 2 y 2 1 128. p 2 36q 2 129. m 2 n 2 4 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 + 3 130. x 2 + 16 131. 8x 3 27y 3 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 Mixed Factoring 121. 3x 3 12x 2 + 36x 122. 6m 3 + 4m 2 2m 123. 3a 3 b 48ab 124. 54x 4 + 2xy 3 137. 3xy 4x 15y + 20 Mixed Factoring 138. 3m 3 3mn 2 139. -6x 3 28x 2 + 10x Algebra II - Polynomials ~7~ NJCTL.org

140. 18a 3 b 50ab 143. 2x 2 y 2 2x 2 y 2xy 2 + 2xy 141. x 4 y + 27xy 142. -12r 3 21r 2 9r Spiral Review 144. Simplify: 145. Simplify: 146. Add: 147. Evaluate, use x = -3, y = 2 8(-4) (2)(-1) + (4) 2 17 2 - (12-4) 2 + 2 2 2 7 + 5 3 5-3x + 2y xy + x Division of Polynomials: Class Work Simplify. 148. 6x3 3x 2 +9x 3x 149. (4a 4 b 3 + 8a 3 b 3 6a 2 b 2 ) (2a 2 b) 150. 6x3 4x 2 +7x+3 3x+1 151. (4a 4 + 8a 3 6a 2 + 3a + 4) (a 1) Algebra II - Polynomials ~8~ NJCTL.org

152. Consider the polynomial function f(x) = 3x 2 + 8x 4. a. Divide f by x 2. b. Find f(2). 153. Consider the polynomial function g(x) = x 3 3x 2 + 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) cm 3 and its height is (t+1) cm. Find the area of the base. (Use V=Bh) Division of Polynomials: Homework Simplify. 156. 16x5 12x 3 +24x 2 4x 2 157. (4a 4 b 3 + 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 3x 2 +9x+2 x+3 160. Consider the polynomial function f(x) = x 3 24. a. Divide f by x 2. b. Find f(2). 161. Consider the polynomial function g(x) = x 3 + 5x 2 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) cm 3. The area of the base, B is (t-1) cm 2. 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. 176. 177. 178. Algebra II - Polynomials ~12~ 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) 186. h(x) = 4 5 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. Algebra II - Polynomials ~13~ NJCTL.org

194. 195. x f(x) x f(x) 196. -2 5 197. -2-4 198. -1 1-1 0 0-1 0 2 1 0 1 1 2 2 2-1 3 1 3-3 4-1 4-1 x f(x) -2 2-1 -3 0-4 1-1 2 2 3 5 4-2 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 f(x) 204. 205. -2 6-1 2 0 1 1 3 2 1 3-1 4 0 x f(x) -2 2-1 4 0 2 1-2 2 0 3 3 4 1 x f(x) -2 4-1 -2 0-3 1-1 2 1 3 3 4 7 Algebra II - Polynomials ~14~ 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. 4 th degree 4 th 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. 216. 3 rd degree 4 th degree 6 th degree Algebra II - Polynomials ~15~ 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. 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. Algebra II - Polynomials ~19~ NJCTL.org

Multiple Choice 1. Simplify the following expression: ( 6p8 q 9 a. b. 3 4pq 3 9 16p 2 q 6 4pq 3 (2p 3 q 4 ) 3) 2 UNIT REVIEW c. 3 d. 16p2 q 6 9 2. The sides of a rectangle are (2x 2 11x +1) ft and (3x 4) ft find the perimeter of the rectangle. a. (2x 2 8x 3) ft b. (4x 2 16x 6) c. (5x 3 11x 3) ft d. (6x 3 41x 2 + 47x 4) ft 2 3. The sides of a rectangle are (2x 2 11x +1) ft and (3x 4) ft find the area of the rectangle. a. (6x 3 41x 2 41x 4) ft 2 b. (6x 3 25x 2 + 47x 4) ft 2 c. (6x 3 41x 2 + 47x 4) ft 2 d. (6x 3 33x 4) ft 2 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) in 2 b. (36x 2 + 4) in 2 c. (36x 2 12x 4) in 2 d. (36x 2 24x + 4) in 2 6. 27w 3 x 5 12w 4 x 3 +24w 3 x 2 6w 2 x 2 is equivalent to which of the following? a. 9wx3 4w 2 x+4w 3 b. 9wx3 2 9wx 3 4w 2 x c. 2w 2 x + 4w 3 + 4w d. 9wx3 +4w 2 x+8w 2 7. (2a 4 6a 2 + 4) (a 2) a. 2a 3 3a 2 b. 2a 3 3a 2 2 c. 2a 3 + 4a 2 2a 4 + 4 a 2 d. 2a 3 + 4a 2 + 2a + 4 + 12 a 2 Algebra II - Polynomials ~20~ NJCTL.org

8. A box has volume of (3x 2 2x 5) cm 3 and a height of (x+1) cm. Find the area of the base of the box. a. (3x + 2) cm 2 b. (3x 2) cm 2 c. (3x + 5) cm 2 d. (3x 5) cm 2 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 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 = (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. x f(x) -2 6-1 A 0 2 1 3 2 1 3-1 4 0 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. j 6 64g 9 h 6 16k 6 9m 2 n 4 64q 6 9p 2 80r 15 t 2 s 8 3 4u 2 v 18 27w 3 y 3 8x 3 z 33 g 16 j 12 81h 12 4k 8 n 8 9m 6 4p 2 q 2 5s 27 t 8 2r 2 8u 8 v 3 3 25w 12 4z 24 13. Yes, 5x 2, degree: 2, monomial/quadratic, 5 14. Yes, -3y 2 +3 4 y, degree: 2, 7 binomial/quadratic, -3 15. Yes, 5a 3-6a+3, degree: 3, trinomial/cubic, 5 16. Not a polynomial function 17. Not a polynomial function 18. 6g 2-4g-7 19. -4t 2 +t+13 20. 7x 5 + 13x 4 + 2x 3 + 6x 1 21. 12x 3 + 6x 2 6x 3 22. Perimeter = (6x 2 +12x-1) inches 23. -8a 3 b+6a 2 b 2 +12a 2 b 24. 35j 4 k 3 +63j 3 k 2-14jk 3 +70jk 2 Answer Key 25. 8x 2-8x-6 26. c 3 +4c 2-3c-12 27. 2m 3-2m 2-17m+15 28. 12f 3 +22f 2-18f+5 29. 12t 4-11t 3 +41t 2-11t+9 30. Area = (30x 2-23x-14) in. 2 31. Area = π(63x 3 +186x 2 +160x+32) m 2 32. Area = (2x 2 +12x+2) ft. 2 33. Areadeck = (4x 2 +100x) ft. 2 34. b 2 +4b+4 35. c 2-2c+1 36. 4d 2 +16de+16e 2 37. 25f 2-81 38. (9x 2 +12x+4) in. 2 39. 32x 5 +320x 4 y+1280x 3 y 2 +2560x 2 y 3 +2560xy 4 +1024y 5 40. 343a 3 +147a 2 b+21ab 2 +b 3 41. 729x 6-5832x 5 z+19440x 4 z 2-34560x 3 z 3 +34560x 2 z 4-18432xz 5 +4096z 6 42. y 4-20y 3 z+150y 2 z 2-500yz 3 +625z 4 43. 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. -2n 2-6n-3 49. -3g 3 +10g 2-5g-11 50. -3x 4 +7x 3 +2x 2-19x-16 51. 6x 3-7x 2 +24x-33 52. Perimeter = (10x 2 +24x-10) inches 53. 12x 3-20x 2-8x 54. -18a 3 b+30a 2 b 2 +42ab Algebra II - Polynomials ~23~ NJCTL.org

55. 16j 5 k 4 +48j 4 k 3-40j 2 k 4 +88j 2 k 3 56. 24x 2 +34x+5 57. 8b 2-40b+18 58. 6c 3 +4c 2-12c-8 59. 6m 3-27m 2 +22m+20 60. 18f 3 +12f 2-13f+4 61. 2p 4 +16p 3 -p 2-40p-10 62. 15t 4-19t 3 +29t 2-15t+6 63. Area = (12x 2-29x+15) in. 2 64. Area = 81x 3-27x+6) m 2 65. Area = (3x 2 +33x+8) in. 2 66. Areadeck = (4x 2 +146x+276) ft. 2 67. 9a 2-1 68. b 2-4b+4 69. c 2-1 70. 9d 2-30de+25e 2 71. 25f 2 +90f+81 72. Area = (16x 2-48xy+36y 2 ) in. 2 73. 64a 6-192a 5 b+240a 4 b 2-160a 3 b 3 +60a 2 b 4-12ab 5 +b 6 74. 27x 3 +54x 2 y+36xy 2 +8y 3 75. 3125y 5-12500y 4 z+20000y 3 z 2-16000y 2 z 3 +6400yz 4-1024z 5 76. a 4 +28a 3 b+294a 2 b 2 +1372ab 3 +2401b 4 77. 3x 2 y(2xy 1) 78. 5p 2 q(2p 3pq q) 79. 7m 3 (n 3 n 2 + 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. 4x 2 y(2x-y) 92. 4mn 3 (2m 2 -m-8) 93. 3pq(-6p 2 q+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 33 107. 56 108. 82 109. (a 1)(a 2 + a + 1) 110. (5x 4y)(5x + 4y) 111. (11a 4b)(11a + 4b) 112. (3x + 2y)(9x 2 + 6xy + 4y 2 ) 113. (ab c)(a 2 b 2 + abc + c 2 ) 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)(9x 2 3xy + y 2 ) 125. x 2 y(x + 10)(x + 2) 126. (y + 3)(y 2 3y + 9) 127. (4m 1)(16m 2 + 4m + 1) 128. (p 6q)(p + 6q) 129. (mn 2)(mn + 2) 130. Not Factorable 131. (2x 3y)(4x 2 + 6xy + 9y 2 ) 132. (2m 5)(3p 1) Algebra II - Polynomials ~24~ NJCTL.org

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)(x 2 3x + 9) 142. -3r(4r + 3)(r + 1) 143. 2xy(x 1)(y 1) 144. 32 145. 227 146. 7 31 35 147. 16 148. 2x 2 -x+3 149. 2a 2 b 2 +4ab 2-3b 150. 2x 2 2x + 3 151. 4a 3 +12a 2 +6a+9 + 13 a 1 152. a. 3x + 14 + 24 b. 24 x 2 153. a. x 2 4x + 10 2 b. -2 x+1 154. Yes, because P(-1) = 0. 155. B = (3t 2-7t + 8-6 t+1 cm.2 156. 4x 3-3x+6 157. a 3 b+2a 2 b- 4a 158. f+ 6 f - 4 f 2 159. 3x 2-12x+45-133 x+3 160. a. x 2 + 2x + 4 16 x 2 161. a. x 2 + 4x 12 + 19 x+1 162. Yes, because P(1) = 0. 163. 4f 2-8f+16-32 f+2 164. height = 4t 2 +t+3+ 5 t 1 cm b. -16 b. 19 165. 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 Algebra II - Polynomials ~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=1 205. 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) Algebra II - Polynomials ~26~ NJCTL.org

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 (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 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 (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 Algebra II - Polynomials ~27~ NJCTL.org

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) = (x 2 3) (x 1 ) (x + 5) 3 239. f(x) = (x 2)(x 3)(x 2 + 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 ) (x 2) 4 243. f(x) = x(x + 1)(x 3) 244. f(x) = x 2 (x + 5)(x 1) 245. f(x) = (x 2 + 4)(x + 5) 3 246. f(x) = x(x 1.5)(x + 1.5) 247. f(x) = x(x 2 1)(x 2 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)(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