Algebra Examples

$x^{8} - 81$

Step 1

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

${(x^{4})}^{2} - 81$

Step 2

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

${(x^{4})}^{2} - 9^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{4}$ and $b = 9$ .

$(x^{4} + 9) (x^{4} - 9)$

Step 4

Simplify.

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Step 4.1

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

$(x^{4} + 9) ({(x^{2})}^{2} - 9)$

Step 4.2

Rewrite $9$ as $3^{2}$ .

$(x^{4} + 9) ({(x^{2})}^{2} - 3^{2})$

Step 4.3

Factor.

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Step 4.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 3$ .

$(x^{4} + 9) ((x^{2} + 3) (x^{2} - 3))$

Step 4.3.2

Remove unnecessary parentheses.

$(x^{4} + 9) (x^{2} + 3) (x^{2} - 3)$