Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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 $b = 3$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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.

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Step 3.1

Cancel the common factor of $x + 3$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x - 3$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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