Precalculus Examples

$x^{5} - 16 x$

Step 1

Factor $x$ out of $x^{5} - 16 x$ .

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Step 1.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 16 x$

Step 1.2

Factor $x$ out of $- 16 x$ .

$x \cdot x^{4} + x \cdot - 16$

Step 1.3

Factor $x$ out of $x \cdot x^{4} + x \cdot - 16$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

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

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

Step 5.1.2

Factor.

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Step 5.1.2.1

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

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

Step 5.1.2.2

Remove unnecessary parentheses.

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

Step 5.2

Remove unnecessary parentheses.

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

Step 6

Multiply the constant in the polynomial $x^{2} + 4$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$x (x^{2} - 4 i^{2}) (x + 2) (x - 2)$

Step 7

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

$x (x^{2} - {(2 i)}^{2}) (x + 2) (x - 2)$

Step 8

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

$x ((x + 2 i) (x - (2 i))) (x + 2) (x - 2)$

Step 9

Factor.

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Step 9.1

Multiply $2$ by $- 1$ .

$x ((x + 2 i) (x - 2 i)) (x + 2) (x - 2)$

Step 9.2

Remove unnecessary parentheses.

$x (x + 2 i) (x - 2 i) (x + 2) (x - 2)$