Algebra Examples

x^{4} + x^{2} + 1

$x^{4} + x^{2} + 1$

Step 1

Rewrite the middle term.

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

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

Step 2

Rearrange terms.

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

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

Step 3

Factor first three terms by perfect square rule.

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

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

Step 4

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} + 1

$a = x^{2} + 1$ and

b = x

$b = x$ .

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

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

Step 5

Simplify.

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Step 5.1

Reorder terms.

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

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

Step 5.2

Reorder terms.

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

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

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

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

x^{4} + x^{2} + 1

$x^{4} + x^{2} + 1$

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