Trigonometry Examples

$x^{5} + 4 x^{3} - 32 x$

Step 1

Factor $x$ out of $x^{5} + 4 x^{3} - 32 x$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Factor $x$ out of $- 32 x$ .

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

Step 1.4

Factor $x$ out of $x \cdot x^{4} + x (4 x^{2})$ .

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

Step 1.5

Factor $x$ out of $x (x^{4} + 4 x^{2}) + x \cdot - 32$ .

$x (x^{4} + 4 x^{2} - 32)$

Step 2

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

$x ({(x^{2})}^{2} + 4 x^{2} - 32)$

Step 3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (u^{2} + 4 u - 32)$

Step 4

Factor $u^{2} + 4 u - 32$ using the AC method.

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Step 4.1

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 $- 32$ and whose sum is $4$ .

$- 4, 8$

Step 4.2

Write the factored form using these integers.

$x ((u - 4) (u + 8))$

Step 5

Replace all occurrences of $u$ with $x^{2}$ .

$x ((x^{2} - 4) (x^{2} + 8))$

Step 6

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

$x ((x^{2} - 2^{2}) (x^{2} + 8))$

Step 7

Factor.

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Step 7.1

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 $b = 2$ .

$x ((x + 2) (x - 2) (x^{2} + 8))$

Step 7.2

Remove unnecessary parentheses.

$x (x + 2) (x - 2) (x^{2} + 8)$