Precalculus Examples

$x^{4} + 9 x^{2} - 400$

Step 1

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

${(x^{2})}^{2} + 9 x^{2} - 400$

Step 2

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

$u^{2} + 9 u - 400$

Step 3

Factor $u^{2} + 9 u - 400$ 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 $- 400$ and whose sum is $9$ .

$- 16, 25$

Step 3.2

Write the factored form using these integers.

$(u - 16) (u + 25)$

Step 4

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

$(x^{2} - 16) (x^{2} + 25)$

Step 5

Rewrite $16$ as $4^{2}$ .

$(x^{2} - 4^{2}) (x^{2} + 25)$

Step 6

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 $b = 4$ .

$(x + 4) (x - 4) (x^{2} + 25)$