Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{1 - x}{1 + x}$ by $\frac{1}{x}$ .

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Move the leading negative in $- \frac{1 - x}{(1 + x) x}$ into the numerator.

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

Step 2.2.2

Factor $x$ out of $(1 + x) x$ .

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

Step 2.2.3

Cancel the common factor.

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

Step 2.2.4

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

Reorder terms.

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

Step 3

Raise $x + 1$ to the power of $1$ .

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

Step 4

Raise $x + 1$ to the power of $1$ .

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

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6

Add $1$ and $1$ .

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

Step 7

Cancel the common factor of $1 - x$ and $x - 1$ .

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Step 7.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 7.2

Factor $- 1$ out of $- x$ .

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

Step 7.3

Factor $- 1$ out of $- 1 (- 1) - (x)$ .

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

Step 7.4

Reorder terms.

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

Step 7.5

Cancel the common factor.

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

Step 7.6

Rewrite the expression.

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

Step 8

Multiply $- 1$ by $- 1$ .

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