Algebra Examples

$\frac{x + 2}{x^{3} + 2 x^{2} - 9 x - 18} \div \frac{3 x + 1}{x^{2} - 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x + 2}{x^{3} + 2 x^{2} - 9 x - 18} \cdot \frac{x^{2} - 9}{3 x + 1}$

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.

$\frac{x + 2}{(x^{3} + 2 x^{2}) - 9 x - 18} \cdot \frac{x^{2} - 9}{3 x + 1}$

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

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

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

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 $b = 3$ .

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

Step 3

Cancel the common factor of $x + 2$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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 $b = 3$ .

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

Step 5

Cancel the common factor of $(x + 3) (x - 3)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{1}{3 x + 1}$