Simplify (x^3-3x^2-4x+12)/(x^3+7x^2-4x-28)

$\frac{x^{3} - 3 x^{2} - 4 x + 12}{x^{3} + 7 x^{2} - 4 x - 28}$

Step 1

Simplify the numerator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x^{3} - 3 x^{2}) - 4 x + 12}{x^{3} + 7 x^{2} - 4 x - 28}$

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

$\frac{x^{2} (x - 3) - 4 (x - 3)}{x^{3} + 7 x^{2} - 4 x - 28}$

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$\frac{(x - 3) (x^{2} - 4)}{x^{3} + 7 x^{2} - 4 x - 28}$

Rewrite $4$ as $2^{2}$ .

$\frac{(x - 3) (x^{2} - 2^{2})}{x^{3} + 7 x^{2} - 4 x - 28}$

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 $b = 2$ .

$\frac{(x - 3) (x + 2) (x - 2)}{x^{3} + 7 x^{2} - 4 x - 28}$

Step 2

Simplify the denominator.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x - 3) (x + 2) (x - 2)}{(x^{3} + 7 x^{2}) - 4 x - 28}$

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

$\frac{(x - 3) (x + 2) (x - 2)}{x^{2} (x + 7) - 4 (x + 7)}$

Factor the polynomial by factoring out the greatest common factor, $x + 7$ .

$\frac{(x - 3) (x + 2) (x - 2)}{(x + 7) (x^{2} - 4)}$

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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