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

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

Simplify the numerator.

Factor $x$ out of $x^{3} - 9 x$ .

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot x^{2} - 9 x}{9 - x^{2}}$

Factor $x$ out of $- 9 x$ .

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

Factor $x$ out of $x \cdot x^{2} + x \cdot - 9$ .

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

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

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

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 $b = 3$ .

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

Simplify the denominator.

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

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

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

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 3$ and $3 + x$ .

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 3$ and $3 - x$ .

Factor $- 1$ out of $x$ .

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

Rewrite $- 3$ as $- 1 (3)$ .

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

Factor $- 1$ out of $- 1 (- x) - 1 (3)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Divide $x \cdot (- 1)$ by $1$ .

$x \cdot (- 1)$

Simplify the expression.

Move $- 1$ to the left of $x$ .

$- 1 x$

Rewrite $- 1 x$ as $- x$ .

$- x$