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• Algebra II - Polynomials ~1~ NJCTL.org

Polynomial Functions

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

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

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

Properties of Exponents: Class Work

Simplify the following expressions.

1. (−4𝑔3ℎ2𝑗−2)−3

2. ( 4𝑘3

3𝑚𝑛2 )

2

3. ( 3p7q3

(2p2q2)3 )

−2

4. (5r3s4t2)(2r3s−3)4

5. (3u2v−4)3(6u4v3)−2

6. ( 8w2x−3y4z5

12w3x−4y5z−6 )

−3

Properties of Exponents: Homework

Simplify the following expressions.

7. (−3𝑔−4ℎ3𝑗−3)−4

8. ( 4𝑘4

6𝑚3𝑛−4 )

2

9. ( 8p7q9

(2p2q2)4 )

−2

10. 4(5r10s12t8)(2r4s−5)−3

11. (6u6v−3)3(9u5v−6)−2

12. ( 6w−3x−4y5z6

15w3x−4y5z−6 )

−2

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

• Algebra II - Polynomials ~2~ NJCTL.org

Operations with Polynomials: Class Work

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

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

13. 2𝑥2 + 3𝑥2

14. 4

7 𝑦 − 3𝑦2 + 3𝑦

15. 5𝑎3 − 2𝑎 − 4𝑎 + 3

16. 6𝑎2

𝑏 − 5𝑎𝑏2 + 2𝑎𝑏2

17. (2𝑥−2 − 4) + (−5𝑥−2 − 3)

Perform the indicated operations.

18. (4g2 − 2) − (3g + 5) + (2g2 − g)

19. (6𝑡 − 3𝑡2 + 4) − (𝑡2 + 5𝑡 − 9)

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

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

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

perimeter of the triangle.

23. −2𝑎(4𝑎2𝑏 − 3𝑎𝑏2 − 6𝑎𝑏)

24. 7𝑗𝑘2(5𝑗3𝑘 + 9𝑗2 − 2𝑘 + 10)

25. (2x − 3)(4x + 2)

26. (𝑐2 − 3)(𝑐 + 4)

27. (m − 3)(2m2 + 4m − 5)

28. (2𝑓 + 5)(6𝑓2 − 4𝑓 + 1)

29. (3t2 − 2t + 9)(4t2 − t + 1)

• Algebra II - Polynomials ~3~ 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 = 𝜋𝑟2ℎ).

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

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

Multiply and simplify:

34. (𝑏 + 2)2

35. (𝑐 − 1)(𝑐 − 1)

36. (2𝑑 + 4𝑒)2

37. (5𝑓 + 9)(5𝑓 − 9)

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

39. (2𝑥 + 4𝑦)5

40. (7𝑎 + 𝑏)3

41. (3𝑥 − 4𝑧)6

42. (𝑦 − 5𝑧)4

• Algebra II - Polynomials ~4~ 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. √2𝑥2 + 0.4𝑥3

44. 4

7𝑦 − 8𝑦2 + 9𝑦

45. 11𝑎4 − 2𝑎3 + 7𝑎2 − 8𝑎 + 9

46. 6𝑎2

11 −

5𝑎

9 + 2

47. (2𝑥 2

3 − 4) + (−5𝑥2 − 3)

Perform the indicated operations:

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

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

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

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

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

rectangle.

53. 4𝑥(3𝑥2 − 5𝑥 − 2)

54. −6𝑎(3𝑎2𝑏 − 5𝑎𝑏2 − 7𝑏)

55. 8𝑗2𝑘3(2𝑗3𝑘 + 6𝑗2 − 5𝑘 + 11)

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

57. (2𝑏 − 9)(4𝑏 − 2)

58. (2𝑐2 − 4)(3𝑐 + 2)

59. (2m − 5)(3m2 − 6m − 4)

60. (3𝑓 + 4)(6𝑓2 − 4𝑓 + 1)

61. (2𝑝2 − 5)(𝑝2 + 8𝑝 + 2)

62. (5t2 − 3t + 6)(3t2 − 2t + 1)

• Algebra II - Polynomials ~5~ 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 𝜋𝑟2ℎ).

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 ~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:

5 – 4 [(-2) – (-2)] 2 3

4 ∙ 4

2

3 2

3

4 ÷ 4

2

3 -2(-6x – 9) + 4

• Algebra II - Polynomials ~7~ NJCTL.org

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

• Algebra II - Polynomials ~8~ NJCTL.org

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 2 2

7 + 5

3

5 -3x + 2y – xy + x

Division of Polynomials: Class Work

Simplify.

148. 6x3−3x2+9x

3x

1