Factor x^4-40x^2+144

$x^{4} - 40 x^{2} + 144$

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

${(x^{2})}^{2} - 40 x^{2} + 144$

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

$u^{2} - 40 u + 144$

Factor $u^{2} - 40 u + 144$ using the AC method.

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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 $144$ and whose sum is $- 40$ .

$- 36, - 4$

Write the factored form using these integers.

$(u - 36) (u - 4)$

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

$(x^{2} - 36) (x^{2} - 4)$

Rewrite $36$ as $6^{2}$ .

$(x^{2} - 6^{2}) (x^{2} - 4)$

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 $b = 6$ .

$(x + 6) (x - 6) (x^{2} - 4)$

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

$(x + 6) (x - 6) (x^{2} - 2^{2})$

Factor.

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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 $b = 2$ .

$(x + 6) (x - 6) ((x + 2) (x - 2))$

Remove unnecessary parentheses.

$(x + 6) (x - 6) (x + 2) (x - 2)$

Factor x^4-40x^2+144

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