Algebra Examples

$\frac{x - 5}{x^{4} - 2 x^{3}} \cdot \frac{x^{2} - 4}{x^{2} - 3 x - 10}$

Step 1

Factor $x^{3}$ out of $x^{4} - 2 x^{3}$ .

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Step 1.1

Factor $x^{3}$ out of $x^{4}$ .

$\frac{x - 5}{x^{3} x - 2 x^{3}} \cdot \frac{x^{2} - 4}{x^{2} - 3 x - 10}$

Step 1.2

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

$\frac{x - 5}{x^{3} x + x^{3} \cdot - 2} \cdot \frac{x^{2} - 4}{x^{2} - 3 x - 10}$

Step 1.3

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

$\frac{x - 5}{x^{3} (x - 2)} \cdot \frac{x^{2} - 4}{x^{2} - 3 x - 10}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{x - 5}{x^{3} (x - 2)} \cdot \frac{x^{2} - 2^{2}}{x^{2} - 3 x - 10}$

Step 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 $b = 2$ .

$\frac{x - 5}{x^{3} (x - 2)} \cdot \frac{(x + 2) (x - 2)}{x^{2} - 3 x - 10}$

Step 3

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 5$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 2$ .

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Step 4.2.1

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

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

Step 4.2.2

Factor $x - 2$ out of $(x + 2) (x - 2)$ .

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 4.3

Multiply $\frac{1}{x^{3}}$ by $\frac{x + 2}{x + 2}$ .

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

Step 4.4

Cancel the common factor of $x + 2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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