Algebra Examples

$x^{4} + 18 x^{2} + 81$

Step 1

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

${(x^{2})}^{2} + 18 x^{2} + 81$

Step 2

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

$u^{2} + 18 u + 81$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $81$ as $9^{2}$ .

$u^{2} + 18 u + 9^{2}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 u = 2 \cdot u \cdot 9$

Step 3.3

Rewrite the polynomial.

$u^{2} + 2 \cdot u \cdot 9 + 9^{2}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 9$ .

${(u + 9)}^{2}$

Step 4

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

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