Algebra Examples

$\frac{x^{3}}{x^{2}} \div \frac{x^{2} - 7 x - 8}{x^{2} - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{3}}{x^{2}} \cdot \frac{x^{2} - 1}{x^{2} - 7 x - 8}$

Step 2

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 2.1

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

$\frac{x^{2} x}{x^{2}} \cdot \frac{x^{2} - 1}{x^{2} - 7 x - 8}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Multiply by $1$ .

$\frac{x^{2} x}{x^{2} \cdot 1} \cdot \frac{x^{2} - 1}{x^{2} - 7 x - 8}$

Step 2.2.2

Cancel the common factor.

$\frac{x^{2} x}{x^{2} \cdot 1} \cdot \frac{x^{2} - 1}{x^{2} - 7 x - 8}$

Step 2.2.3

Rewrite the expression.

$\frac{x}{1} \cdot \frac{x^{2} - 1}{x^{2} - 7 x - 8}$

Step 2.2.4

Divide $x$ by $1$ .

$x \frac{x^{2} - 1}{x^{2} - 7 x - 8}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$x \frac{x^{2} - 1^{2}}{x^{2} - 7 x - 8}$

Step 3.2

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 $b = 1$ .

$x \frac{(x + 1) (x - 1)}{x^{2} - 7 x - 8}$

Step 4

Factor $x^{2} - 7 x - 8$ using the AC method.

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Step 4.1

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 $- 8$ and whose sum is $- 7$ .

$- 8, 1$

Step 4.2

Write the factored form using these integers.

$x \frac{(x + 1) (x - 1)}{(x - 8) (x + 1)}$

Step 5

Cancel the common factor of $x + 1$ .

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Step 5.1

Cancel the common factor.

$x \frac{(x + 1) (x - 1)}{(x - 8) (x + 1)}$

Step 5.2

Rewrite the expression.

$x \frac{x - 1}{x - 8}$

Step 6

Combine $x$ and $\frac{x - 1}{x - 8}$ .

$\frac{x (x - 1)}{x - 8}$