Algebra Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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 $b = 1$ .

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

Step 1.1.3

Simplify.

Tap for more steps...

Step 1.1.3.1

Multiply $x$ by $1$ .

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

Step 1.1.3.2

One to any power is one.

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

Step 1.2

Simplify the denominator.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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 $b = 1$ .

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

Step 1.3

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Multiply $- 1$ by $1$ .

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

Step 2

Find the common denominator.

Tap for more steps...

Step 2.1

Write $- x$ as a fraction with denominator $1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Write $- 1$ as a fraction with denominator $1$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.2.1

Move $x$ .

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

Step 4.2.2

Multiply $x$ by $x$ .

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

Step 4.3

Multiply $- 1$ by $1$ .

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Multiply $- 1$ by $1$ .

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

Step 5

Simplify by adding terms.

Tap for more steps...

Step 5.1

Combine the opposite terms in $x^{2} + x + 1 - x^{2} - x - x - 1$ .

Tap for more steps...

Step 5.1.1

Subtract $x^{2}$ from $x^{2}$ .

$\frac{x + 1 + 0 - x - x - 1}{x + 1}$

Step 5.1.2

Add $x + 1$ and $0$ .

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

Step 5.1.3

Subtract $x$ from $x$ .

$\frac{0 + 1 - x - 1}{x + 1}$

Step 5.1.4

Add $0$ and $1$ .

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

Step 5.1.5

Subtract $1$ from $1$ .

$\frac{- x + 0}{x + 1}$

Step 5.1.6

Add $- x$ and $0$ .

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

Step 5.2

Move the negative in front of the fraction.

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