Algebra Examples

$\frac{x^{2} - 4}{x^{2} + 2 x - 8} \cdot \frac{8 x + 32}{x^{4} + x^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} - 2^{2}}{x^{2} + 2 x - 8} \cdot \frac{8 x + 32}{x^{4} + x^{3}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{(x + 2) (x - 2)}{x^{2} + 2 x - 8} \cdot \frac{8 x + 32}{x^{4} + x^{3}}$

Step 2

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 2) (x - 2)}{(x - 2) (x + 4)} \cdot \frac{8 x + 32}{x^{4} + x^{3}}$

Step 3

Simplify terms.

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Step 3.1

Factor $8$ out of $8 x + 32$ .

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Step 3.1.1

Factor $8$ out of $8 x$ .

$\frac{(x + 2) (x - 2)}{(x - 2) (x + 4)} \cdot \frac{8 (x) + 32}{x^{4} + x^{3}}$

Step 3.1.2

Factor $8$ out of $32$ .

$\frac{(x + 2) (x - 2)}{(x - 2) (x + 4)} \cdot \frac{8 x + 8 \cdot 4}{x^{4} + x^{3}}$

Step 3.1.3

Factor $8$ out of $8 x + 8 \cdot 4$ .

$\frac{(x + 2) (x - 2)}{(x - 2) (x + 4)} \cdot \frac{8 (x + 4)}{x^{4} + x^{3}}$

Step 3.2

Factor $x^{3}$ out of $x^{4} + x^{3}$ .

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Step 3.2.1

Factor $x^{3}$ out of $x^{4}$ .

$\frac{(x + 2) (x - 2)}{(x - 2) (x + 4)} \cdot \frac{8 (x + 4)}{x^{3} x + x^{3}}$

Step 3.2.2

Multiply by $1$ .

$\frac{(x + 2) (x - 2)}{(x - 2) (x + 4)} \cdot \frac{8 (x + 4)}{x^{3} x + x^{3} \cdot 1}$

Step 3.2.3

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot 1$ .

$\frac{(x + 2) (x - 2)}{(x - 2) (x + 4)} \cdot \frac{8 (x + 4)}{x^{3} (x + 1)}$

Step 3.3

Combine.

$\frac{(x + 2) (x - 2) (8 (x + 4))}{(x - 2) (x + 4) (x^{3} (x + 1))}$

Step 3.4

Cancel the common factor of $x - 2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{(x + 2) (x - 2) (8 (x + 4))}{(x - 2) (x + 4) (x^{3} (x + 1))}$

Step 3.4.2

Rewrite the expression.

$\frac{(x + 2) (8 (x + 4))}{(x + 4) (x^{3} (x + 1))}$

Step 3.5

Cancel the common factor of $x + 4$ .

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Step 3.5.1

Cancel the common factor.

$\frac{(x + 2) (8 (x + 4))}{(x + 4) (x^{3} (x + 1))}$

Step 3.5.2

Rewrite the expression.

$\frac{(x + 2) \cdot (8)}{x^{3} (x + 1)}$

Step 3.6

Move $8$ to the left of $x + 2$ .

$\frac{8 (x + 2)}{x^{3} (x + 1)}$