Write as a Set of Linear Factors f(x)=2x^3+3x^2+18x+27

$f (x) = 2 x^{3} + 3 x^{2} + 18 x + 27$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$f (x) = (2 x^{3} + 3 x^{2}) + 18 x + 27$

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

$f (x) = x^{2} (2 x + 3) + 9 (2 x + 3)$

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$f (x) = (2 x + 3) (x^{2} + 9)$

Factor over the complex numbers.

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

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

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

$f (x) = (2 x + 3) (x^{2} - {(3 i)}^{2})$

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 $i = 3 i$ .

$f (x) = (2 x + 3) ((x + 3 i) (x - (3 i)))$

Factor.

Multiply $3$ by $- 1$ .

$f (x) = (2 x + 3) ((x + 3 i) (x - 3 i))$

Remove unnecessary parentheses.

$f (x) = (2 x + 3) (x + 3 i) (x - 3 i)$