Algebra Examples

$\frac{x^{4} - 9 x^{2}}{x^{3} - 3 x^{2}} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x^{2}$ out of $x^{4} - 9 x^{2}$ .

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Step 1.1.1

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

$\frac{x^{2} x^{2} - 9 x^{2}}{x^{3} - 3 x^{2}} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 1.1.2

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

$\frac{x^{2} x^{2} + x^{2} \cdot - 9}{x^{3} - 3 x^{2}} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 1.1.3

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

$\frac{x^{2} (x^{2} - 9)}{x^{3} - 3 x^{2}} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 1.2

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

$\frac{x^{2} (x^{2} - 3^{2})}{x^{3} - 3 x^{2}} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 1.3

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 $b = 3$ .

$\frac{x^{2} (x + 3) (x - 3)}{x^{3} - 3 x^{2}} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 2

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

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Step 2.1

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

$\frac{x^{2} (x + 3) (x - 3)}{x^{2} x - 3 x^{2}} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 2.2

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

$\frac{x^{2} (x + 3) (x - 3)}{x^{2} x + x^{2} \cdot - 3} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 3$ .

$\frac{x^{2} (x + 3) (x - 3)}{x^{2} (x - 3)} \cdot \frac{x + 5}{x^{2} + 2 x - 15}$

Step 3

Factor $x^{2} + 2 x - 15$ using the AC method.

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Step 3.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 3.2

Write the factored form using these integers.

$\frac{x^{2} (x + 3) (x - 3)}{x^{2} (x - 3)} \cdot \frac{x + 5}{(x - 3) (x + 5)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 3$ .

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Step 4.1.1

Factor $x - 3$ out of $x^{2} (x + 3) (x - 3)$ .

$\frac{(x - 3) (x^{2} (x + 3))}{x^{2} (x - 3)} \cdot \frac{x + 5}{(x - 3) (x + 5)}$

Step 4.1.2

Cancel the common factor.

$\frac{(x - 3) (x^{2} (x + 3))}{x^{2} (x - 3)} \cdot \frac{x + 5}{(x - 3) (x + 5)}$

Step 4.1.3

Rewrite the expression.

$\frac{x^{2} (x + 3)}{x^{2} (x - 3)} \cdot \frac{x + 5}{x + 5}$

Step 4.2

Multiply $\frac{x^{2} (x + 3)}{x^{2} (x - 3)}$ by $\frac{x + 5}{x + 5}$ .

$\frac{x^{2} (x + 3) (x + 5)}{x^{2} (x - 3) (x + 5)}$

Step 4.3

Cancel the common factor of $x^{2}$ .

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Step 4.3.1

Cancel the common factor.

$\frac{x^{2} (x + 3) (x + 5)}{x^{2} (x - 3) (x + 5)}$

Step 4.3.2

Rewrite the expression.

$\frac{(x + 3) (x + 5)}{(x - 3) (x + 5)}$

Step 4.4

Cancel the common factor of $x + 5$ .

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Step 4.4.1

Cancel the common factor.

$\frac{(x + 3) (x + 5)}{(x - 3) (x + 5)}$

Step 4.4.2

Rewrite the expression.

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