Write as a Set of Linear Factors p(x)=x^3+x^2+9x+9

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

Step 1

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$p (x) = x^{2} (x + 1) + 9 (x + 1)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$p (x) = (x + 1) (x^{2} + 9)$

Step 3

Factor over the complex numbers.

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

$p (x) = (x + 1) (x^{2} - 9 i^{2})$

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

$p (x) = (x + 1) (x^{2} - {(3 i)}^{2})$

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 $i = 3 i$ .

$p (x) = (x + 1) ((x + 3 i) (x - (3 i)))$

Factor.

Multiply $3$ by $- 1$ .

$p (x) = (x + 1) ((x + 3 i) (x - 3 i))$

Remove unnecessary parentheses.

$p (x) = (x + 1) (x + 3 i) (x - 3 i)$