Precalculus Examples

$p (x) = x^{4} - x^{2} - 2$

Step 1

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

$p (x) = {(x^{2})}^{2} - x^{2} - 2$

Step 2

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

$p (x) = u^{2} - u - 2$

Step 3

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

$- 2, 1$

Step 3.2

Write the factored form using these integers.

$p (x) = (u - 2) (u + 1)$

Step 4

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

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