Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(x - 1) (x^{2} - 1)$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 1$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

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

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

Step 4.2.1.2

Add $1$ and $2$ .

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

Step 4.2.2

Move $- 1$ to the left of $x$ .

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

Step 4.2.3

Rewrite $- 1 x$ as $- x$ .

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

Step 4.2.4

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

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

Step 4.2.5

Multiply $- 1$ by $- 1$ .

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

Step 4.3

Multiply $x^{3} - x - x^{2} + 1$ by $1$ .

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

Step 4.4

Add $- x^{2}$ and $x^{2}$ .

$\frac{x^{3} - x + 0 + 1 - 1}{(x - 1) (x^{2} - 1) \cdot 1 + x - 1}$

Step 4.5

Add $x^{3}$ and $0$ .

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

Step 4.6

Subtract $1$ from $1$ .

$\frac{x^{3} - x + 0}{(x - 1) (x^{2} - 1) \cdot 1 + x - 1}$

Step 4.7

Add $x^{3} - x$ and $0$ .

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

Step 4.8

Rewrite $x^{3} - x$ in a factored form.

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Step 4.8.1

Factor $x$ out of $x^{3} - x$ .

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Step 4.8.1.1

Factor $x$ out of $x^{3}$ .

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

Step 4.8.1.2

Factor $x$ out of $- x$ .

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

Step 4.8.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 1$ .

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

Step 4.8.2

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

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

Step 4.8.3

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 $b = 1$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

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

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

Step 5.2.1.2

Add $1$ and $2$ .

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

Step 5.2.2

Move $- 1$ to the left of $x$ .

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

Step 5.2.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.2.4

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

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

Step 5.2.5

Multiply $- 1$ by $- 1$ .

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

Step 5.3

Multiply $x^{3} - x - x^{2} + 1$ by $1$ .

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

Step 5.4

Add $- x$ and $x$ .

$\frac{x (x + 1) (x - 1)}{x^{3} - x^{2} + 1 + 0 - 1}$

Step 5.5

Add $x^{3}$ and $0$ .

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

Step 5.6

Subtract $1$ from $1$ .

$\frac{x (x + 1) (x - 1)}{x^{3} - x^{2} + 0}$

Step 5.7

Add $x^{3} - x^{2}$ and $0$ .

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

Step 5.8

Factor $x^{2}$ out of $x^{3} - x^{2}$ .

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Step 5.8.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 5.8.2

Factor $x^{2}$ out of $- x^{2}$ .

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

Step 5.8.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 1$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $x$ and $x^{2}$ .

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Step 6.1.1

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

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

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

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

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

Step 6.1.2.2

Cancel the common factor.

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

Step 6.1.2.3

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x - 1$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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