Algebra Examples

$\frac{x - 2}{4 x^{3}} \cdot \frac{x^{2} - 1}{x^{2} - x - 2}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x - 2}{4 x^{3}} \cdot \frac{x^{2} - 1^{2}}{x^{2} - x - 2}$

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 $b = 1$ .

$\frac{x - 2}{4 x^{3}} \cdot \frac{(x + 1) (x - 1)}{x^{2} - x - 2}$

Step 2

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

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Step 2.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 2.2

Write the factored form using these integers.

$\frac{x - 2}{4 x^{3}} \cdot \frac{(x + 1) (x - 1)}{(x - 2) (x + 1)}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x - 2$ .

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Step 3.1.1

Cancel the common factor.

$\frac{x - 2}{4 x^{3}} \cdot \frac{(x + 1) (x - 1)}{(x - 2) (x + 1)}$

Step 3.1.2

Rewrite the expression.

$\frac{1}{4 x^{3}} \cdot \frac{(x + 1) (x - 1)}{x + 1}$

Step 3.2

Multiply $\frac{1}{4 x^{3}}$ by $\frac{(x + 1) (x - 1)}{x + 1}$ .

$\frac{(x + 1) (x - 1)}{4 x^{3} (x + 1)}$

Step 3.3

Cancel the common factor of $x + 1$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(x + 1) (x - 1)}{4 x^{3} (x + 1)}$

Step 3.3.2

Rewrite the expression.

$\frac{x - 1}{4 x^{3}}$