Algebra Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 2.1.2

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

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

Step 2.1.3

Cancel the common factor.

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

Step 2.1.4

Rewrite the expression.

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

Step 2.2

Multiply $- 1$ by $3$ .

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

Step 2.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.3.1

Apply the product rule to $- \frac{1}{x}$ .

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

Step 2.3.2

Apply the product rule to $\frac{1}{x}$ .

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

Step 2.4

Raise $- 1$ to the power of $2$ .

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

Step 2.5

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

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

Step 2.6

Cancel the common factor of $x$ .

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Step 2.6.1

Factor $x$ out of $3 x$ .

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

Step 2.6.2

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

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

Step 2.6.3

Cancel the common factor.

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

Step 2.6.4

Rewrite the expression.

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

Step 2.7

Combine $3$ and $\frac{1^{2}}{x}$ .

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

Step 2.8

One to any power is one.

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

Step 2.9

Multiply $3$ by $1$ .

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

Step 2.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.10.1

Apply the product rule to $- \frac{1}{x}$ .

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

Step 2.10.2

Apply the product rule to $\frac{1}{x}$ .

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

Step 2.11

Raise $- 1$ to the power of $3$ .

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

Step 2.12

One to any power is one.

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