Precalculus Examples

$f (x) = x^{4} - 16$

Step 1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f (x) = {(x^{2})}^{2} - 16$

Step 2

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

$f (x) = {(x^{2})}^{2} - 4^{2}$

Step 3

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

$f (x) = (x^{2} + 4) (x^{2} - 4)$

Step 4

Simplify.

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Step 4.1

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

$f (x) = (x^{2} + 4) (x^{2} - 2^{2})$

Step 4.2

Factor.

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Step 4.2.1

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 $b = 2$ .

$f (x) = (x^{2} + 4) ((x + 2) (x - 2))$

Step 4.2.2

Remove unnecessary parentheses.

$f (x) = (x^{2} + 4) (x + 2) (x - 2)$