Algebra Examples

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

Step 1

To write $\frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 2

To write $- \frac{1}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 3

Write each expression with a common denominator of $(x - 1) (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(x - 1) (x + 1)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $- 1$ .

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

Step 5.3

Subtract $x$ from $x$ .

$(x^{2} - 1) \cdot (\frac{0 + 1 + 1}{(x + 1) (x - 1)} + 1)$

Step 5.4

Add $0$ and $1$ .

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

Expand $(x^{2} - 1) (\frac{2}{(x + 1) (x - 1)} + 1)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Apply the distributive property.

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

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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

Step 7.1.2

Move $2$ to the left of $x^{2}$ .

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

Step 7.1.3

Multiply $x^{2}$ by $1$ .

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

Step 7.1.4

Rewrite $- 1 \frac{2}{(x + 1) (x - 1)}$ as $- \frac{2}{(x + 1) (x - 1)}$ .

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

Step 7.1.5

Multiply $- 1$ by $1$ .

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

Step 7.2

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

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