Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Add $- 1$ and $3$ .

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

Step 6

To write $\frac{x + 2}{(x + 1) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 7

To write $- \frac{2}{{(x + 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 8

Write each expression with a common denominator of $(x - 1) {(x + 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{x + 2}{(x + 1) (x - 1)}$ by $\frac{x + 1}{x + 1}$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Expand $(x + 2) (x + 1)$ using the FOIL Method.

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Step 10.1.1

Apply the distributive property.

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

Step 10.1.2

Apply the distributive property.

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

Step 10.1.3

Apply the distributive property.

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

Step 10.2

Simplify and combine like terms.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Multiply $x$ by $x$ .

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

Step 10.2.1.2

Multiply $x$ by $1$ .

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

Step 10.2.1.3

Multiply $2$ by $1$ .

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

Step 10.2.2

Add $x$ and $2 x$ .

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

Step 10.3

Apply the distributive property.

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

Step 10.4

Multiply $- 2$ by $- 1$ .

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

Step 10.5

Subtract $2 x$ from $3 x$ .

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

Step 10.6

Add $2$ and $2$ .

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