Factor y=x^4-5x^2+4

$y = x^{4} - 5 x^{2} + 4$

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

$y = {(x^{2})}^{2} - 5 x^{2} + 4$

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

$y = u^{2} - 5 u + 4$

Factor $u^{2} - 5 u + 4$ using the AC method.

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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Write the factored form using these integers.

$y = (u - 4) (u - 1)$

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

$y = (x^{2} - 4) (x^{2} - 1)$

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

$y = (x^{2} - 2^{2}) (x^{2} - 1)$

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 $b = 2$ .

$y = (x + 2) (x - 2) (x^{2} - 1)$

Rewrite $1$ as $1^{2}$ .

$y = (x + 2) (x - 2) (x^{2} - 1^{2})$

Factor.

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 $b = 1$ .

$y = (x + 2) (x - 2) ((x + 1) (x - 1))$

Remove unnecessary parentheses.

$y = (x + 2) (x - 2) (x + 1) (x - 1)$

Factor y=x^4-5x^2+4

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