Algebra Examples

$f (x) = x^{4} + x^{2} - 12$

Step 1

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

$f (x) = {(x^{2})}^{2} + x^{2} - 12$

Step 2

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

$f (x) = u^{2} + u - 12$

Step 3

Factor $u^{2} + u - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 3.2

Write the factored form using these integers.

$f (x) = (u - 3) (u + 4)$

Step 4

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

$f (x) = (x^{2} - 3) (x^{2} + 4)$