Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Combine the opposite terms in $x^{2} + 1 + x - 1 + x + 1$ .

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Step 3.2.1

Subtract $1$ from $1$ .

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

Step 3.2.2

Add $x^{2} + x$ and $0$ .

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

Step 3.3

Add $x$ and $x$ .

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

Step 4

Factor using the perfect square rule.

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Step 4.1

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

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

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 4.3

Rewrite the polynomial.

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

Step 4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 5

Cancel the common factor of ${(x + 1)}^{2}$ and $x + 1$ .

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Step 5.1

Factor $x + 1$ out of ${(x + 1)}^{2}$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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