Algebra Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

Multiply $- 1$ by $3$ .

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

Step 2.1.2

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 2.1.3

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

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

Step 2.2

Move $- 3$ to the left of $x^{2}$ .

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

Step 2.3

Move the negative in front of the fraction.

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

Step 2.4

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

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Step 2.4.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 2.4.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

One to any power is one.

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

Step 2.8

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

$x^{3} - \frac{3 x^{2}}{2} + 3 x \frac{1}{4} + {(- \frac{1}{2})}^{3}$

Step 2.9

Multiply $3 x \frac{1}{4}$ .

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Step 2.9.1

Combine $3$ and $\frac{1}{4}$ .

$x^{3} - \frac{3 x^{2}}{2} + x \frac{3}{4} + {(- \frac{1}{2})}^{3}$

Step 2.9.2

Combine $x$ and $\frac{3}{4}$ .

$x^{3} - \frac{3 x^{2}}{2} + \frac{x \cdot 3}{4} + {(- \frac{1}{2})}^{3}$

Step 2.10

Move $3$ to the left of $x$ .

$x^{3} - \frac{3 x^{2}}{2} + \frac{3 \cdot x}{4} + {(- \frac{1}{2})}^{3}$

Step 2.11

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

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Step 2.11.1

Apply the product rule to $- \frac{1}{2}$ .

$x^{3} - \frac{3 x^{2}}{2} + \frac{3 x}{4} + {(- 1)}^{3} {(\frac{1}{2})}^{3}$

Step 2.11.2

Apply the product rule to $\frac{1}{2}$ .

$x^{3} - \frac{3 x^{2}}{2} + \frac{3 x}{4} + {(- 1)}^{3} \frac{1^{3}}{2^{3}}$

Step 2.12

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

$x^{3} - \frac{3 x^{2}}{2} + \frac{3 x}{4} - \frac{1^{3}}{2^{3}}$

Step 2.13

One to any power is one.

$x^{3} - \frac{3 x^{2}}{2} + \frac{3 x}{4} - \frac{1}{2^{3}}$

Step 2.14

Raise $2$ to the power of $3$ .

$x^{3} - \frac{3 x^{2}}{2} + \frac{3 x}{4} - \frac{1}{8}$