Simplify (x^2-9)/(x^3+x^2-9x-9)

$\frac{x^{2} - 9}{x^{3} + x^{2} - 9 x - 9}$

Step 1

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$\frac{x^{2} - 3^{2}}{x^{3} + x^{2} - 9 x - 9}$

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

$\frac{(x + 3) (x - 3)}{x^{3} + x^{2} - 9 x - 9}$

Step 2

Simplify the denominator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x + 3) (x - 3)}{(x^{3} + x^{2}) - 9 x - 9}$

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

$\frac{(x + 3) (x - 3)}{x^{2} (x + 1) - 9 (x + 1)}$

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

$\frac{(x + 3) (x - 3)}{(x + 1) (x^{2} - 9)}$

Rewrite $9$ as $3^{2}$ .

$\frac{(x + 3) (x - 3)}{(x + 1) (x^{2} - 3^{2})}$

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

$\frac{(x + 3) (x - 3)}{(x + 1) (x + 3) (x - 3)}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{(x + 3) (x - 3)}{(x + 1) (x + 3) (x - 3)}$

Rewrite the expression.

$\frac{x - 3}{(x + 1) (x - 3)}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{x - 3}{(x + 1) (x - 3)}$

Rewrite the expression.

$\frac{1}{x + 1}$