Precalculus Examples

$f (x) = x^{4} + 10 x^{2} + 9$

Step 1

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

$f (x) = {(x^{2})}^{2} + 10 x^{2} + 9$

Step 2

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

$f (x) = u^{2} + 10 u + 9$

Step 3

Factor $u^{2} + 10 u + 9$ 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 $9$ and whose sum is $10$ .

$1, 9$

Step 3.2

Write the factored form using these integers.

$f (x) = (u + 1) (u + 9)$

Step 4

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

$f (x) = (x^{2} + 1) (x^{2} + 9)$