Write as a Set of Linear Factors f(x)=x^4+81x^2

$f (x) = x^{4} + 81 x^{2}$

Step 1

Factor $x^{2}$ out of $x^{4} + 81 x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

$f (x) = x^{2} x^{2} + 81 x^{2}$

Factor $x^{2}$ out of $81 x^{2}$ .

$f (x) = x^{2} x^{2} + x^{2} \cdot 81$

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot 81$ .

$f (x) = x^{2} (x^{2} + 81)$

Step 2

Factor over the complex numbers.

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

$f (x) = x^{2} (x^{2} - 81 i^{2})$

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

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

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 $i = 9 i$ .

$f (x) = x^{2} ((x + 9 i) (x - (9 i)))$

Factor.

Multiply $9$ by $- 1$ .

$f (x) = x^{2} ((x + 9 i) (x - 9 i))$

Remove unnecessary parentheses.

$f (x) = x^{2} (x + 9 i) (x - 9 i)$