Algebra Examples

$x^{8} + x^{4} + 1$

Step 1

Rewrite the middle term.

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

Step 2

Rearrange terms.

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

Step 3

Factor first three terms by perfect square rule.

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

Step 4

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

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

Step 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 $b = x^{2}$ .

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

Step 6

Simplify.

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Step 6.1

Reorder terms.

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

Step 6.2

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

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Step 6.2.1

Rewrite the middle term.

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

Step 6.2.2

Rearrange terms.

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

Step 6.2.3

Factor first three terms by perfect square rule.

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

Step 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 $b = x$ .

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

Step 6.2.5

Simplify.

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Step 6.2.5.1

Reorder terms.

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

Step 6.2.5.2

Reorder terms.

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

Step 6.3

Reorder terms.

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