Precalculus Examples

$x^{5} + 4 x^{3} + 8 x^{2} + 32$

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.

$(x^{5} + 4 x^{3}) + 8 x^{2} + 32$

Step 1.2

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

$x^{3} (x^{2} + 4) + 8 (x^{2} + 4)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $x^{2} + 4$ .

$(x^{2} + 4) (x^{3} + 8)$

Step 3

Rewrite $8$ as $2^{3}$ .

$(x^{2} + 4) (x^{3} + 2^{3})$

Step 4

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

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Multiply $2$ by $- 1$ .

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

Step 5.1.2

Raise $2$ to the power of $2$ .

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

Step 5.2

Remove unnecessary parentheses.

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