Algebra Examples

$x^{5} + x^{3} - x^{2} - 1$

Step 1

Factor out the greatest common factor from each group.

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Step 1.1

Group the first two terms and the last two terms.

$(x^{5} + x^{3}) - x^{2} - 1$

Step 1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2

Factor the polynomial by factoring out the greatest common factor, $x^{2} + 1$ .

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

Step 3

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

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

Step 4

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 $a = x$ and $b = 1$ .

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Multiply $x$ by $1$ .

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

Step 5.1.2

One to any power is one.

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

Step 5.2

Remove unnecessary parentheses.

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