Algebra Examples

$x^{6} - 1$

Step 1

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

${(x^{2})}^{3} - 1$

Step 2

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

${(x^{2})}^{3} - 1^{3}$

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 $a = x^{2}$ and $b = 1$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $x^{2}$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Multiply $2$ by $2$ .

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

Step 6

One to any power is one.

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

Step 7

Factor.

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Step 7.1

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

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Step 7.1.1

Rewrite the middle term.

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

Step 7.1.2

Rearrange terms.

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

Step 7.1.3

Factor first three terms by perfect square rule.

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

Step 7.1.4

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 $b = x$ .

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

Step 7.1.5

Simplify.

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Step 7.1.5.1

Reorder terms.

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

Step 7.1.5.2

Reorder terms.

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

Step 7.2

Remove unnecessary parentheses.

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