Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Factor $x$ out of $- x$ .

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

Step 1.1.3

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

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

Step 1.2

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

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

Step 1.3

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 $b = 1$ .

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

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $x - 1$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Divide $x (x + 1)$ by $1$ .

$x (x + 1)$

Step 2.2

Apply the distributive property.

$x \cdot x + x \cdot 1$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $x$ by $x$ .

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

Step 2.3.2

Multiply $x$ by $1$ .

$x^{2} + x$