Algebra 2 CP
Name__________________
Chapter 5
Test Review
Simplify the following expressions using the exponent rules.
1. 10 5  10 4  10
 a3
4.  5
b

7. 5 y
10.




4

3 2
4m 4

2. 2a 2 b

 3x 6
5.  10
 2y
5
y y

3n 1
 6m 1 m  2
8.
5




 
3. 4c 5
3
 c11
 
6. 5 0
3
 c 
9.   4 
d 
x3 y6
x 5 y 2
 qr 2 s 
11.  4 
 3r 
2
3
12.
2
x 2 y 3 2x 4
 3
2
y
Describe the following and fill in the missing information
13. g x   x 3  7 x 2  2 x 5  4
14. hx   2 x 3  7 x 4  8
Polynomial Function:
Polynomial Function:
Standard Form:
Standard Form:
Degree:
Degree:
Type:
Type:
Leading Coefficient:
Leading Coefficient:
End Behavior:
End Behavior:
Perform the indicated operations. Describe the end behavior of the simplified polynomial.
15. h(x) = (2x3 + 5x2 – 7x + 4) + (-3x2 + x3 – 4x)
16. g(x) = (3x3 – 4x2 + 3x – 5) – (x2 + 4x – 8)
17. f(x) = (-x4 + x2 – x – x3 + 1) + (x2 – 2x3 + 4x – 1)
18. f(x) = (4x5 + 3x4 – 5x + 1) – (x3 + 2x4 – x5 + 1)
Perform the indicated operations.
19. (3x – 2)(x2 + 4x – 7)
20. (3x – 5)3
21. (2y + 3z)3
22. (x2 + 6x – 2)(-x2 + 3)
23. (3x3 – 14x2 + 16x – 22) ÷ (x – 4)
24. (6x4 + 7x2 + 4x – 17) ÷ (3x2 – 3x + 2)
25. (6x2 – 5x + 9) ÷ (2x – 1)
26. (2x3 + 3x5 + 1 – 5x) ÷ (x – 1)
Factor the following polynomials completely. Choose whatever method you like.
27. 8x3 + 27
28. x4 + 5x2 – 6
29. x3 – 3x2 – 4x + 12
30. 40v3 – 625
31. 2b4 + 14b3 – 16b – 112
32. 6y6 – 5y3 - 4
Evaluate the following polynomials using a) Direct Substitution and b) Synthetic Substitution
33. Find p(-2) give p(x) = 2x3 – x – 8
34. Evaluate f(x) = -3x4 + 6x2 + 2x – 10 when x = 3
List all possible rational zeros the function. Then find all real zeros.
35. f(x) = x3 + x2 – 22x – 40
36. f(x) = 4x4 – 8x3 – 19x2 + 23x – 6
37. g(x) = 2x4 + 5x3 – 5x2 – 5x + 3
38. h(x) = 2x4 + 3x3 – 6x2 – 6x + 4
Find all the zeros of the function.
39. f(x) = x3 – 8x2 – 15x + 54
40. g(x) = x3 + 8x2 – 7x – 56
41. g(x) = x4 – 9x3 + 23x2 – 81x + 126
42. h(x) = 2x4 + x3 + x2 + x - 1
Use the graphs below to answer the following questions:
a) How many real zeros does this function have? b) Describe the end behavior; c) How many turning
points are there? d) Approximate the local max and min.
43.
State the maximum number of turns in the graph of the function.
44. f(x) = x4 + 2x2 + 4
45. g(x) = -3x3 + x2 – x + 5
46. f(x) = 2x6 + 1
Graph the following functions.
47. f(x) = (x – 3)(x + 2)(x + 1)
48. g ( x) 
5
x  12 ( x  1)( x  4)
6
Find the following information for the following polynomials: a) Zeros; b) Relative Maximum; c)
Relative Minimum; d) Graph. Round answers to the nearest tenth
49. f(x) = x3 – 5x2 + 3x + 4
50. g(x) = x4 + 3x3 – x2 – 6x + 2
Write a polynomial function f of least degree that has integer coefficients, a leading coefficient of 1,
and the given zeros.
51. -1, 3, 4
52. 4, 3i
53. 2, 1 3
54. 8, 2 + i
Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the
function.
55. h(x) = x3 – 4x2 + 5x + 9
56. f(x) = x5 – 6x4 – 3x3 + 7x2 – 8x + 1
Write a cubic function whose graph passes through the points.
57. (-2, 0), (0, 0), (1, 0), (2, 1)
58. (-4, 0), (-1, 0), (3, 0), (2, -2)
59. (-3, 0), (-1, 10), (0, 0), (4, 0)
60. (-2, 0), (-1, 0), (0, -8), (2, 0)
61. (-3, 0), (-1, 0), (3, 0), (0, 3)
62. A rectangular prism has edges of lengths x, x + 2, and 2x – 3 inches. The volume of the prism is 1040
cubic inches. Write a polynomial equation that models the prism’s volume. What are the prism’s dimensions.
63. You are going to do a math story problem that you’ll think is unrealistic. You are building a cage for a
classroom fish. You want the length of the cage to be six times the height, and the width to be three times the
height. The sides should be one foot thick. Because the cage will be on a desk, it does not need a bottom.
Somehow, the water will not spill out of the bottom. It also doesn’t have a lid, just in case you want to go
fishing. What should the outer dimensions of the cage be if it is to hold 4 cubic feet of water?
64. The expression for the area of a rectangle is x2 + 10x + 21. If the width is x + 3, find an expression for the
length.
Answers
1.
10 10
2. 2 5 a 10 b 5
3.
6.
1
7.
8.
11.
33 r 6
25
6
12. x
q3s3
4 2 c 21
a 12
4.
y8
9.
x2
b 20
1
5
3
f ( x)   as x  
f ( x)   as x  
f ( x)   as x  
f ( x)   as x  
2
10
x4
g ( x)  3x 3  5 x 2  x  3
16. g ( x)   as x  
g ( x)   as x  
f ( x)  5 x 5  x 4  x 3  5 x
18. f ( x)   as x  
23. 3x  2 x  8 
2m 7
n
2
15. h( x)   as x  
h( x)   as x  
17. f ( x)   as x  
3
2
20. 27 x  135 x  225 x  125
2 3 y 30
13. Yes; g ( x)  2 x  x  7 x  4 ; Degree: 5; 5th degree;
h( x)  3x 3  2 x 2  11x  4
f ( x)   x 4  3x 3  2 x 2  3x
3 3 x 18
10. 
c2d 8
Leading Coefficient: 2;
14. Not a Polynomial Function
5.
21. 8 y  36 y z  54 yz  27 z
3
2
2
2
24. 2 x  2 x  3 
9 x  23
3x  3x  2
2
3
2
19. 3x  10 x  29 x  14
3
4
3
2
22.  x  6 x  5 x  18 x  6
25. 3x  1 
8
2x 1
1
26. 3x 4  3x 3  5 x 2  5 x 
x 1
27. (2 x  3)(4 x  6 x  9)
28. ( x  6)( x  1)( x  1)
29. ( x  3)( x  2)( x  2)
30. 5(2v  5)(4v  10v  25)
31. 2(b  7)(b  2)(b  2b  4)
2
2
2
2
32. (2 y  1)(3 y  4)
3
3
33. p(-2) = -22
35.
p
: 1, 2, 4, 5, 8, 10, 20, 40 ; x = 5, -4, -2
q
37.
p
3
: 1, 3, ;
q
2
1
p
1 3 1 3
: 1, 2, 3, 6, , , , ; x  3,  2,
2
q
2 2 4 4
1
p
1
38.
:  1, 2, 4, ; x   2, ,  2
2
q
2
1
41. x  2, 7,  3i
42. x   1,  i
2
36.
x = -3, -1, 1, ½
40. x  8,  7
39. x = -3, 9, 2
43. a) 3; b) f ( x)   as x  ;
34. f(3) = -193
f ( x)   as x   ; c) 2; d) Max (-3, 2.5) Min (0.5, -2.5)
51. f ( x)  x 3  6 x 2  5 x  12
52. f ( x)  x  4 x  9 x  36
53. f ( x)  x  4 x  2 x  4
54. f ( x)  x  12 x  37 x  40
55. Pos: 2, 0; Neg: 1; Im: 2, 0
56. Pos: 4, 2, 0; Neg: 1; Im: 0, 2, 4
57. f ( x) 
3
2
59. V  2 x  x  6 x  1040
60. 1 foot by 3 feet by 6 feet
61. Length: x + 7
27. (2 x  3)(4 x  6 x  9)
28. ( x  6)( x  1)( x  1)
29. ( x  3)( x  2)( x  2)
30. 5(2v  5)(4v  10v  25)
31. 2(b  7)(b  2)(b  2b  4)
32. (2 y  1)(3 y  4)
33. p(-2) = -22
34. f(3) = -193
44. 3
45. 2
3
46. 5
2
26. 3x 4  3x 3  5 x 2  5 x 
3
3
1
x 1
2
1
x( x  2)( x  1)
8
58. f ( x) 
2
2
3
35.
p
: 1, 2, 4, 5, 8, 10, 20, 40 ; x = 5, -4, -2
q
37.
p
3 1
: 1, 3, , ; x = -3, -1, 1, ½
q
2 2
43. a) 3; b) f ( x)   as x  ;
2
1
( x  4)( x  1)( x  3)
9
2
2
p
1 3 1 3
2 3
: 1, 2, 3, 6, , , , ; x  3,  2,
4
q
2 2 4 4
1
p
1
38.
:  1, 2, 4, ; x   2, ,  2
2
q
2
1
41. x  2, 7,  3i
42. x   1,  i
2
36.
40. x  8,  7
39. x = -3, 9, 2
3
f ( x)   as x   ; c) Odd; d) 2; e) Max (-3, 2.5) Min (0.5, -2.5)
51. f ( x)  x 3  6 x 2  5 x  12
52. f ( x)  x  4 x  9 x  36
53. f ( x)  x  4 x  2 x  4
54. f ( x)  x  12 x  37 x  40
55. Pos: 2, 0; Neg: 1; Im: 2, 0
56. Pos: 4, 2, 0; Neg: 1; Im: 0, 2, 4
57. f ( x) 
44. 3
45. 2
3
58. f ( x) 
46. 5
2
1
( x  4)( x  1)( x  3)
9
3
2
1
x( x  2)( x  1)
8
59. f ( x)  x( x  4)( x  3)
1
( x  3)( x  3)( x  1)
3
60. f ( x)  2( x  2)( x  2)( x  1)
61. f ( x)  
62. V ( x)  2 x  x  6 x  1040
63. 1 foot by 3 feet by 6 feet
3
2
3
64. Length: x + 7
2