Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{3}$ .

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

Step 1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $x$ by $1$ .

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

Step 1.3.2

One to any power is one.

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

Step 2

To write $- \frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + x + 1}{x^{2} + x + 1}$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $1$ .

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

Step 4.3

Subtract $x$ from $x$ .

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

Step 4.4

Add $2$ and $0$ .

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

Step 4.5

Subtract $1$ from $2$ .

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

Step 4.6

Rewrite $- x^{2} + 1$ in a factored form.

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Step 4.6.1

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

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

Step 4.6.2

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 4.6.3

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

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $1 - x$ and $x - 1$ .

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Step 5.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.1.2

Factor $- 1$ out of $- x$ .

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

Step 5.1.3

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

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

Step 5.1.4

Reorder terms.

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

Step 5.1.5

Cancel the common factor.

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

Step 5.1.6

Rewrite the expression.

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

Step 5.2

Simplify the expression.

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Step 5.2.1

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

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

Step 5.2.2

Move the negative in front of the fraction.

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