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BENCHMARK 2 Practice/Review (Algebra 2)
CA Standards:
(3.0) Students are adept at operations on polynomials, including long division.
(4.0) Students factor polynomials representing the difference of squares, perfect square
trinomials, and difference/sum of two cubes.
(7.0) Students perform operations on rational expressions with polynomials in the
numerator and denominator. Simplify rational expressions with negative exponents.
(15.0) Students determine whether a specific algebraic statement involving rational
expressions, radical expressions,… is sometimes true, always true, or never true.
NOTES: Special Factoring Patterns

Difference of Two Squares: (a2 – b2) = (a + b)(a – b)
Ex. 4x2 – 9 = (2x)2 – (3)2
= (2x + 3)(2x – 3)

Sum of Two Cubes: (a3 + b3) = (a + b)(a2 – ab + b2)
Ex. (x3 + 8) = (x)3 + (2)3
= (x + 2)(x2 – 2x + 4)

Difference of Two Cubes: (a3 – b3) = (a – b)(a2 + ab + b2)
Ex. (8x3 – 1) = (2x)3 – (1)3
= (2x – 1)(4x2 + 2x +1)
I. Simplify the expression.
1. (6xy3)2 =
2. 7x-10y4 =
3. (5/4) -2 =
4. 3x2y-1  10x2y
2x
3y-3
II. Perform the indicated operation.
5. (x – 3)(x3 – 2x2 + 5x – 12)
6. (7x3 – 9x + 2) + (5x3 + 9x)
7. (-3x3 + x – 11) – (4x3 + x2 – x)
8. (10x3 – 4x2 + 3x) – (x3 – x2 + 1)
9. (x4 -3x3 + 8x2 – 2)/(x + 2)
10. (10x3 + 27x2 + 14x + 5)/(x2 + 2x)
III. Find all the zeros of the function.
11. f(x) = 2x3 – 5x2 – 4x + 3
12. f(x) = x4 – 25
13. f(x) = x3 + 11x2 + x + 11
14. Use synthetic division to decide
which of the numbers 2, -2, 3,
and -3 are zeros of
f(x) = 2x3 + 4x2 – 18x – 36.
IV. Choose the best answer.
15. What is the quotient of (x + 2)  (x2 – 9x – 22)/(x2 – 121)?
A) x + 11
B) (x+11)/(x+2)
C) (x+2)/(x+11)
D) (x+2)/(x-11)
E) x + 2
16. What is the difference of
8x – 3
x2 + 2x – 35
-
7 ?
x2 - 25
A) 2(4x2 + 15x – 17)
(x2 + 2x – 35)(x+5)
B) 2(4x2 + 15x + 32)
(x2 + 2x – 35)(x+5)
C) 2(4x2 + 15x + 17)
(x2 + 2x – 35)(x+5)
D) 2(4x2 + 15x – 32)
(x2 + 2x – 35)(x + 5)
17. What is the simplified form of the following complex fraction?
10
x+1
1 + 3
2
x+1
A) 20x
x+7
B) 20
x+7
C) 10
x+7
D) 10(x+7)
x+1
E) 20
V. Simplify the following.
19. x – 1 – x – 4
x–2 x+1
18. 5 + 7
6x 18x
21.
4x – 8 
x2 – 3x + 2
3x – 6
x–1
20.
3x
+
x2 – 10x + 21
5
x–3
22. x + 4  (x2 +3x – 10)
x2 – 25
VI. Match the following polynomials with their factors.
23. 3x2 + 11x + 6
A. (5x – 2)(25x2 + 10x + 4)
24. x3 – 4x2 + 4x – 16
B. 2(x + 3)(x2 - 3x + 9)
25. 125x3 – 216
C. 2x3(x+2)(x-2)(x2+4)
26. 2x7 – 32x3
D. 2x(x+4)(x-4)
27. 2x5 + 4x4 – 4x3 - 8x2
E. (3x+2)(x+3)
28. 2x3 – 32x
F. (x2+4)(x-4)
29. 125x3 – 8
G. 2x2(x2 – 2)(x + 2)
30. 2x3 + 54
H. (5x -6)(25x2 + 30x + 36)