Precalculus Examples

$\frac{16 - x^{2}}{x^{3} - 64}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $16$ as $4^{2}$ .

$\frac{4^{2} - x^{2}}{x^{3} - 64}$

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 $a = 4$ and $b = x$ .

$\frac{(4 + x) (4 - x)}{x^{3} - 64}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $64$ as $4^{3}$ .

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

Step 2.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 4$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Move $4$ to the left of $x$ .

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

Step 2.3.2

Raise $4$ to the power of $2$ .

$\frac{(4 + x) (4 - x)}{(x - 4) (x^{2} + 4 x + 16)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $4 - x$ and $x - 4$ .

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Step 3.1.1

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{(4 + x) (- 1 (- 4) - x)}{(x - 4) (x^{2} + 4 x + 16)}$

Step 3.1.2

Factor $- 1$ out of $- x$ .

$\frac{(4 + x) (- 1 (- 4) - (x))}{(x - 4) (x^{2} + 4 x + 16)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- 4) - (x)$ .

$\frac{(4 + x) (- 1 (- 4 + x))}{(x - 4) (x^{2} + 4 x + 16)}$

Step 3.1.4

Reorder terms.

$\frac{(4 + x) (- 1 (x - 4))}{(x - 4) (x^{2} + 4 x + 16)}$

Step 3.1.5

Cancel the common factor.

$\frac{(4 + x) (- 1 (x - 4))}{(x - 4) (x^{2} + 4 x + 16)}$

Step 3.1.6

Rewrite the expression.

$\frac{(4 + x) \cdot (- 1)}{x^{2} + 4 x + 16}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move $- 1$ to the left of $4 + x$ .

$\frac{- 1 \cdot (4 + x)}{x^{2} + 4 x + 16}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{4 + x}{x^{2} + 4 x + 16}$