Factor over the Complex Numbers x^6+9

$x^{6} + 9$

Step 1

Multiply the constant in the polynomial $x^{6} + 9$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$x^{6} - 9 i^{2}$

Step 2

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

${(x^{3})}^{2} - 9 i^{2}$

Step 3

Rewrite $9 i^{2}$ as ${(3 i)}^{2}$ .

${(x^{3})}^{2} - {(3 i)}^{2}$

Step 4

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

$(x^{3} + 3 i) (x^{3} - (3 i))$

Step 5

Multiply $3$ by $- 1$ .

$(x^{3} + 3 i) (x^{3} - 3 i)$

Factor over the Complex Numbers x^6+9

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