Simplify (x^3+4x^2-21x)/(x^3-9x)

$\frac{x^{3} + 4 x^{2} - 21 x}{x^{3} - 9 x}$

Simplify the numerator.

Factor $x$ out of $x^{3} + 4 x^{2} - 21 x$ .

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

$\frac{x \cdot x^{2} + 4 x^{2} - 21 x}{x^{3} - 9 x}$

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

$\frac{x \cdot x^{2} + x (4 x) - 21 x}{x^{3} - 9 x}$

Factor $x$ out of $- 21 x$ .

$\frac{x \cdot x^{2} + x (4 x) + x \cdot - 21}{x^{3} - 9 x}$

Factor $x$ out of $x \cdot x^{2} + x (4 x)$ .

$\frac{x (x^{2} + 4 x) + x \cdot - 21}{x^{3} - 9 x}$

Factor $x$ out of $x (x^{2} + 4 x) + x \cdot - 21$ .

$\frac{x (x^{2} + 4 x - 21)}{x^{3} - 9 x}$

Factor $x^{2} + 4 x - 21$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 21$ and whose sum is $4$ .

$- 3, 7$

Write the factored form using these integers.

$\frac{x ((x - 3) (x + 7))}{x^{3} - 9 x}$

$\frac{x (x - 3) (x + 7)}{x^{3} - 9 x}$

Simplify the denominator.

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

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

$\frac{x (x - 3) (x + 7)}{x \cdot x^{2} - 9 x}$

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

$\frac{x (x - 3) (x + 7)}{x \cdot x^{2} + x \cdot - 9}$

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

$\frac{x (x - 3) (x + 7)}{x (x^{2} - 9)}$

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

$\frac{x (x - 3) (x + 7)}{x (x^{2} - 3^{2})}$

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 $b = 3$ .

$\frac{x (x - 3) (x + 7)}{x (x + 3) (x - 3)}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x (x - 3) (x + 7)}{x (x + 3) (x - 3)}$

Rewrite the expression.

$\frac{(x - 3) (x + 7)}{(x + 3) (x - 3)}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{(x - 3) (x + 7)}{(x + 3) (x - 3)}$

Rewrite the expression.

$\frac{x + 7}{x + 3}$

Simplify (x^3+4x^2-21x)/(x^3-9x)

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