Algebra Examples

$x^{12} - 1$

Step 1

Rewrite $x^{12}$ as ${(x^{4})}^{3}$ .

${(x^{4})}^{3} - 1$

Step 2

Rewrite $1$ as $1^{3}$ .

${(x^{4})}^{3} - 1^{3}$

Step 3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x^{4}$ and $b = 1$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Simplify.

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Step 4.4.1

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

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

Step 4.4.2

Factor.

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Step 4.4.2.1

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

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

Step 4.4.2.2

Remove unnecessary parentheses.

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

Step 4.5

Multiply $x^{4}$ by $1$ .

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

Step 5

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.2

Multiply $4$ by $2$ .

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

Step 6

One to any power is one.

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

Step 7

Factor.

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Step 7.1

Rewrite $x^{8} + x^{4} + 1$ in a factored form.

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Step 7.1.1

Rewrite the middle term.

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

Step 7.1.2

Rearrange terms.

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

Step 7.1.3

Factor first three terms by perfect square rule.

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

Step 7.1.4

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

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

Step 7.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{4} + 1$ and $b = x^{2}$ .

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

Step 7.1.6

Simplify.

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Step 7.1.6.1

Reorder terms.

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

Step 7.1.6.2

Rewrite $x^{4} + x^{2} + 1$ in a factored form.

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Step 7.1.6.2.1

Rewrite the middle term.

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

Step 7.1.6.2.2

Rearrange terms.

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

Step 7.1.6.2.3

Factor first three terms by perfect square rule.

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

Step 7.1.6.2.4

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

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

Step 7.1.6.2.5

Simplify.

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Step 7.1.6.2.5.1

Reorder terms.

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

Step 7.1.6.2.5.2

Reorder terms.

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

Step 7.1.6.3

Reorder terms.

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

Step 7.2

Remove unnecessary parentheses.

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