Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Multiply $x$ by $x$ .

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

Step 4.2

Multiply $x$ by $1$ .

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

Step 4.3

Reorder terms.

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

Step 5

Factor using the perfect square rule.

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Step 5.1

Rearrange terms.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Rewrite the polynomial.

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

Step 5.9

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

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