Factor over the Complex Numbers x^2+81

$x^{2} + 81$

Multiply the constant in the polynomial $x^{2} + 81$ by $- i^{2}$ where $i^{2}$ is equal to $- 1$ .

$x^{2} - 81 i^{2}$

Rewrite $81 i^{2}$ as ${(9 i)}^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = x$ and $i = 9 i$ .

$(x + 9 i) (x - (9 i))$

Multiply $9$ by $- 1$ .

$(x + 9 i) (x - 9 i)$