This document provides a study guide for Chapter 6 of Algebra 2 covering evaluating expressions using laws of exponents, determining if functions are polynomials, describing polynomial end behavior, factoring polynomials, solving polynomial equations, dividing polynomials using long and synthetic division, finding zeros of polynomials, writing polynomials given their zeros, and graphing polynomials. It includes 79 problems to work through involving these topics.

1. Algebra 2
Chapter 6 Study Guide
Evaluate each expression using laws of exponents.
3
4 2  5
1. (3 ) 2.   3. (-2)-3(-2)9
 8
3
 a2  rs 2
4.  b −3  5. 6. (-y2)5y2y-12
  (rs −1 )3
3
 a 2b  xy 9 −7 ψ y10 20x14
7.  a −1β5  8. γ 9. g
  3y −2 21ξ5 2x 3 xy 6
State whether the function is a polynomial. If so, name the (a) leading coefficient, (b)
degree, and (c) constant term of each polynomial:
1
10. f (x) = ξ2 − 3ξ4 − 7 11. f (x) = ξ3 + 3ξ
2
12. f (x) = 6 ξ2 + 2 ξ−1 + ξ 13. f (x) = −0.5 ξ + π ξ2 − 2
14. f (x) = 2 ξ2 − ξ−2 15. f (x) = −0.8 ξ3 + ξ4 − 5
16. What two things do you need to look at to determine a graph’s end behavior?
Describe the end behavior of the following polynomial functions.
17. f (x) = 5 ξ4 18. f (x) = − ξ6 + 2 ξ3 − ξ
19. f (x) = 3ξ5 − 4 ξ2 + 3 19. f (x) = − ξ3 + 10 ξ
Simplify.
( 5x + ξ − 7 ) + ( −3ξ − 6 ξ − 1) ( x + 2ξ + 8) + (2 ξ − 9)
2 2 4 3 4
20. 21.
3
( 3x + 8 ξ − ξ − 5 ) − ( 5 ξ − ξ + 17 ) 23.
2 3 2
( 2x − 3)
3
22.
( 9x − 12 ξ + ξ − 8 ) − ( 3ξ − 12 ξ − ξ)
4 3 2 4 3
24.
25. ( x + 2 ) ( 5 ξ2 + 3ξ − 1) 26. ( x − 2 ) ( ξ − 1) ( ξ + 3)
2
27. ( 3x − 2 ) ( 3ξ + 2 ) 28. ( 5x + 2 )
Factor each expression completely.
29. x3 + 27 30. 16x5 – 250x2 31. 64x4 – 27x
32. x3 – 2x2 – 9x + 18 33. x3 + x2 + x + 1 34. 2x3 – x2 + 2x – 1
4
35. 16x – 1 36. x4 + 3x2 + 2 37. 6x5 – 51x3 – 27x
Solve each equation.
38. x3 – 3x2 = 0 39. 2x3 – 6x2 = 0 40. x4 + 7x3 – 8x – 56 = 0
41. x3 + 2x2 – x = 2 42. x3 + 8x2 = – 16x 43. 3x4 + 3x3 = 6x2 + 6x

2. Use long division to divide the following polynomials.
44. (2x 4 + 3ξ3 + 5 ξ − 1) ÷ ( ξ2 − 2 ξ + 2) 45. (x 4 + 2 ξ2 − ξ + 5) ÷ ( ξ2 − ξ + 1)
46. (10x 3 + 27 ξ2 + 14 ξ + 5) ÷ ( ξ2 + 2 ξ) 47. (5x 4 + 14 ξ3 + 9 ξ) ÷ ( ξ2 + 3ξ)
Use synthetic division to divide the following polynomials.
48. (x 3 − 7 ξ − 6) ÷ ( ξ − 2) 49. (4 x 2 + 5 ξ − 4) ÷ ( ξ − 2)
50. (3x 2 − 10 ξ) ÷ ( ξ − 6) 51. (x 4 − 6 ξ3 − 40 ξ + 33) ÷ ( ξ − 7)
Find all the zeros of each polynomial.
52. f (x) = ξ3 + 2 ξ2 − 11ξ − 12 53. f (x) = ξ3 − 4 ξ2 − 11ξ + 30
54. f (x) = ξ3 − ξ2 − 9 ξ + 9 55. f (x) = ξ3 − 7 ξ2 + 10 ξ + 6
56. f (x) = 10 ξ4 − 3ξ3 − 29 ξ2 + 5 ξ + 12 57. f (x) = ξ3 − 7 ξ2 + 2 ξ + 40
58. f (x) = ξ5 − 2 ξ4 + 8 ξ2 − 13ξ + 6 59. f (x) = ξ4 + 3ξ3 − 8 ξ2 − 22 ξ − 24
Write a polynomial f of least degree that has real coefficients, a leading coefficient of 1,
and the following zeros.
60. 1, 2, 1 + i 61. 5, 2i, and -2i 62. 1 and -2 + i
63. 3 – i and 5i 64. -2, -2, 3, -4i 65. 4, 4, 2 + i
66. If 3 is a zero of polynomial f, then __________ is a factor of f(x).
67. If (x + 2) is a factor of f(x), then ________ is an x-intercept of the graph of f(x).
Graph each polynomial.
1
68. f (x) = ( ξ + 2)( ξ − 1)2 69. f (x) = −2( ξ2 − 9)( ξ + 4)
4
70. f (x) = ξ3 + 2 ξ2 − 5 ξ + 1 71. f (x) = 2 ξ4 − 5 ξ3 − 4 ξ2 − 6
Write an equation to model the polynomial given the following points.
72. (-1, 0)(-2, 0)(0, 0)(1, -3) 73. (1, 0)(3, 0)(-2, 0)(2, 1)
74. (3, 0)(2, 0)(-1, 0)(1, 4) 75. (0, 0)(-3, 0)(5, 0)(-2, 3)
Show that the nth-order differences for the given function of degree n are nonzero and
constant.
76. f (x) = ξ2 − 3ξ + 7 77. f (x) = − ξ3 + 3ξ2 − 2 ξ − 3
78. f (x) = 2 ξ4 − 20 79. f (x) = 3ξ3 − 5 ξ2 − 2
Use finite differences and a system of equations to find the polynomial function that fits
the data.
x 1 2 3 4 5 6
f(x) -4 0 10 26 48 76