Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the denominator.

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Step 2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + 1$ and $b = x$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Add $x$ and $x$ .

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

Step 2.2.2

Subtract $x$ from $x$ .

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

Step 2.2.3

Add $0$ and $1$ .

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

Step 3

Multiply $2 x + 1$ by $1$ .

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

Step 4

Multiply the numerator and denominator of the fraction by $1 + x$ .

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Step 4.1

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

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

Step 4.2

Combine.

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

Step 5

Apply the distributive property.

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

Step 6

Simplify by cancelling.

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Step 6.1

Cancel the common factor of $1 + x$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $1 + x$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

Factor $x$ out of $x^{1}$ .

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

Step 7.1.4

Factor $x$ out of $x (1 + x) + x \cdot 1$ .

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

Step 7.2

Add $1$ and $1$ .

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

Step 8

Simplify the denominator.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $x$ by $1$ .

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

Step 8.3

Multiply $x$ by $x$ .

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

Step 8.4

Reorder terms.

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

Step 9

Simplify terms.

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Step 9.1

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

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Step 9.1.1

Cancel the common factor.

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

Step 9.1.2

Rewrite the expression.

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

Step 9.2

Combine $x$ and $\frac{1}{2 x + 1}$ .

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

Step 9.3

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

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