Algebra 2 Review worksheet featuring 20 multiple-choice questions covering factoring, solving equations, and simplifying rational expressions.

Let's solve each of these problems step by step.

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1. Which of the following is the factored form of $ x^3 - 125 $?

This is a difference of cubes:

$$
x^3 - 125 = x^3 - 5^3
$$

The formula for difference of cubes is:
$$
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
$$

So,
$$
x^3 - 125 = (x - 5)(x^2 + 5x + 25)
$$

✔ Answer: C
(x - 5)(x² + 5x + 25)

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2. What is the solution set to $ x^2 = 2x - 5 $?

Bring all terms to one side:
$$
x^2 - 2x + 5 = 0
$$

Use the quadratic formula:
$$
x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(5)}}{2(1)} = \frac{2 \pm \sqrt{4 - 20}}{2} = \frac{2 \pm \sqrt{-16}}{2}
$$
$$
= \frac{2 \pm 4i}{2} = 1 \pm 2i
$$

✔ Answer: D
{1 ± 2i}

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3. Which is a factor of $ 3x^2 + 10x - 8 = 0 $?

We need to factor this quadratic.

Look for two numbers that multiply to $ 3 \times (-8) = -24 $ and add to 10.

Try: 12 and -2 → $ 12 \times (-2) = -24 $, $ 12 + (-2) = 10 $

Split the middle term:
$$
3x^2 + 12x - 2x - 8 = 3x(x + 4) -2(x + 4) = (3x - 2)(x + 4)
$$

So the factors are $ (3x - 2) $ and $ (x + 4) $

Check options:
- A: (x - 4) → no
- B: (x + 6) → no
- C: (2x - 3) → no
- D: (3x - 2) → ✔ Yes!

✔ Answer: D
(3x - 2)

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4. Simplify: $ \frac{n - 6}{n^2 - 13n + 42} $

Factor the denominator:
$$
n^2 - 13n + 42 = (n - 6)(n - 7)
$$

So:
$$
\frac{n - 6}{(n - 6)(n - 7)} = \frac{1}{n - 7}, \quad \text{for } n \ne 6
$$

✔ Answer: D
1/(n - 7)

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5. Simplify the rational function: $ \frac{x + 1}{3x^2 + 3x} $

Factor denominator:
$$
3x^2 + 3x = 3x(x + 1)
$$

So:
$$
\frac{x + 1}{3x(x + 1)} = \frac{1}{3x}, \quad \text{for } x \ne -1
$$

✔ Answer: D
1/(3x)

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6. Divide: $ \frac{6}{k - 4} \div \frac{k + 6}{k^2 + 2k - 24} $

First, rewrite as multiplication:
$$
\frac{6}{k - 4} \cdot \frac{k^2 + 2k - 24}{k + 6}
$$

Factor $ k^2 + 2k - 24 $:
Find two numbers that multiply to -24 and add to 2 → 6 and -4
$$
k^2 + 2k - 24 = (k + 6)(k - 4)
$$

Now substitute:
$$
\frac{6}{k - 4} \cdot \frac{(k + 6)(k - 4)}{k + 6}
$$

Cancel common terms:
- $ k - 4 $ cancels
- $ k + 6 $ cancels

Left with:
$$
6
$$

✔ Answer: C
6

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✔ Final Answers:

1. C
2. D
3. D
4. D
5. D
6. C

Let me know if you'd like explanations for any others or help with more questions!