Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

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

Step 1.1.2

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 $b = 1$ .

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

Step 1.2

Cancel the common factor of $x + 1$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Factor using the perfect square rule.

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Step 1.3.1

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

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

Step 1.3.2

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 1.3.3

Rewrite the polynomial.

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

Step 1.3.4

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

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

Step 1.4

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

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Step 1.4.1

Multiply by $1$ .

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

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

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $1$ and $1$ .

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