Factor x^16-1

$x^{16} - 1$

Step 1

Rewrite $x^{16}$ as ${(x^{8})}^{2}$ .

${(x^{8})}^{2} - 1$

Step 2

Rewrite $1$ as $1^{2}$ .

${(x^{8})}^{2} - 1^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{8}$ and $b = 1$ .

$(x^{8} + 1) (x^{8} - 1)$

Step 4

Simplify.

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

$(x^{8} + 1) ({(x^{4})}^{2} - 1)$

Rewrite $1$ as $1^{2}$ .

$(x^{8} + 1) ({(x^{4})}^{2} - 1^{2})$

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 $b = 1$ .

$(x^{8} + 1) ((x^{4} + 1) (x^{4} - 1))$

Factor.

Simplify.

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

$(x^{8} + 1) ((x^{4} + 1) ({(x^{2})}^{2} - 1))$

Rewrite $1$ as $1^{2}$ .

$(x^{8} + 1) ((x^{4} + 1) ({(x^{2})}^{2} - 1^{2}))$

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 $b = 1$ .

$(x^{8} + 1) ((x^{4} + 1) ((x^{2} + 1) (x^{2} - 1)))$

Factor.

Simplify.

Rewrite $1$ as $1^{2}$ .

$(x^{8} + 1) ((x^{4} + 1) ((x^{2} + 1) (x^{2} - 1^{2})))$

Factor.

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 $b = 1$ .

$(x^{8} + 1) ((x^{4} + 1) ((x^{2} + 1) ((x + 1) (x - 1))))$

Remove unnecessary parentheses.

$(x^{8} + 1) ((x^{4} + 1) ((x^{2} + 1) (x + 1) (x - 1)))$

Remove unnecessary parentheses.

$(x^{8} + 1) ((x^{4} + 1) (x^{2} + 1) (x + 1) (x - 1))$

Remove unnecessary parentheses.

$(x^{8} + 1) (x^{4} + 1) (x^{2} + 1) (x + 1) (x - 1)$

Factor x^16-1

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