Algebra Examples

$x^{4} - 9 x^{2} + 20$

Step 1

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

${(x^{2})}^{2} - 9 x^{2} + 20$

Step 2

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

$u^{2} - 9 u + 20$

Step 3

Factor $u^{2} - 9 u + 20$ using the AC method.

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Step 3.1

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 $20$ and whose sum is $- 9$ .

$- 5, - 4$

Step 3.2

Write the factored form using these integers.

$(u - 5) (u - 4)$

Step 4

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

$(x^{2} - 5) (x^{2} - 4)$

Step 5

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

$(x^{2} - 5) (x^{2} - 2^{2})$

Step 6

Factor.

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Step 6.1

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 $b = 2$ .

$(x^{2} - 5) ((x + 2) (x - 2))$

Step 6.2

Remove unnecessary parentheses.

$(x^{2} - 5) (x + 2) (x - 2)$