Algebra Examples

x^{8} - 1

$x^{8} - 1$

Step 1

Rewrite

x^{8}

$x^{8}$ as

{(x^{4})}^{2}

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

{(x^{4})}^{2} - 1

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

Step 2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

{(x^{4})}^{2} - 1^{2}

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x^{4}

$a = x^{4}$ and

b = 1

$b = 1$ .

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

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

Step 4

Simplify.

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Step 4.1

Rewrite

x^{4}

$x^{4}$ as

{(x^{2})}^{2}

${(x^{2})}^{2}$ .

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

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

Step 4.2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x^{2}

$a = x^{2}$ and

b = 1

$b = 1$ .

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

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

Step 4.4

Factor.

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Step 4.4.1

Simplify.

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Step 4.4.1.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 4.4.1.2

Factor.

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Step 4.4.1.2.1

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 1

$b = 1$ .

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

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

Step 4.4.1.2.2

Remove unnecessary parentheses.

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

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

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

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

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

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

Step 4.4.2

Remove unnecessary parentheses.

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

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

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

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

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

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