Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $x$ by $1$ .

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

Step 1.3.2

One to any power is one.

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Factor $- 1$ out of $x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Reorder terms.

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

Step 2.1.5

Cancel the common factor.

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

Step 2.1.6

Divide $- 1 (x^{2} + x + 1)$ by $1$ .

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

Step 2.2

Rewrite $- 1 (x^{2} + x + 1)$ as $- (x^{2} + x + 1)$ .

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

Step 2.3

Apply the distributive property.

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

Step 2.4

Multiply $- 1$ by $1$ .

$- x^{2} - x - 1$