Simplify (x^3+2x^2-9x-18)/(x^3-x^2-6x)

$\frac{x^{3} + 2 x^{2} - 9 x - 18}{x^{3} - x^{2} - 6 x}$

Step 1

Simplify the numerator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x^{3} + 2 x^{2}) - 9 x - 18}{x^{3} - x^{2} - 6 x}$

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

$\frac{x^{2} (x + 2) - 9 (x + 2)}{x^{3} - x^{2} - 6 x}$

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

$\frac{(x + 2) (x^{2} - 9)}{x^{3} - x^{2} - 6 x}$

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

$\frac{(x + 2) (x^{2} - 3^{2})}{x^{3} - x^{2} - 6 x}$

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 $b = 3$ .

$\frac{(x + 2) (x + 3) (x - 3)}{x^{3} - x^{2} - 6 x}$

Step 2

Simplify the denominator.

Factor $x$ out of $x^{3} - x^{2} - 6 x$ .

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

$\frac{(x + 2) (x + 3) (x - 3)}{x \cdot x^{2} - x^{2} - 6 x}$

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

$\frac{(x + 2) (x + 3) (x - 3)}{x \cdot x^{2} + x (- x) - 6 x}$

Factor $x$ out of $- 6 x$ .

$\frac{(x + 2) (x + 3) (x - 3)}{x \cdot x^{2} + x (- x) + x \cdot - 6}$

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

$\frac{(x + 2) (x + 3) (x - 3)}{x (x^{2} - x) + x \cdot - 6}$

Factor $x$ out of $x (x^{2} - x) + x \cdot - 6$ .

$\frac{(x + 2) (x + 3) (x - 3)}{x (x^{2} - x - 6)}$

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

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

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (x^3+2x^2-9x-18)/(x^3-x^2-6x)

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