Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $x$ by $1$ .

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

Step 1.3.2

One to any power is one.

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

Step 2

Simplify terms.

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Step 2.1

Factor $x$ out of $x^{3} + x^{2} + x$ .

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Step 2.1.1

Factor $x$ out of $x^{3}$ .

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

Step 2.1.2

Factor $x$ out of $x^{2}$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Factor $x$ out of $x \cdot x^{2} + x \cdot x$ .

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

Step 2.1.6

Factor $x$ out of $x (x^{2} + x) + x \cdot 1$ .

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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