Factor x^6+2x^4-16x^2-32

$x^{6} + 2 x^{4} - 16 x^{2} - 32$

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{6} + 2 x^{4}) - 16 x^{2} - 32$

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

$x^{4} (x^{2} + 2) - 16 (x^{2} + 2)$

Step 2

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

$(x^{2} + 2) (x^{4} - 16)$

Step 3

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

$(x^{2} + 2) ({(x^{2})}^{2} - 16)$

Step 4

Rewrite $16$ as $4^{2}$ .

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

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 4$ .

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

Step 6

Factor.

Simplify.

Rewrite $4$ as $2^{2}$ .

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

Factor.

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 $b = 2$ .

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

Remove unnecessary parentheses.

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

Remove unnecessary parentheses.

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

Factor x^6+2x^4-16x^2-32

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