Algebra Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(x)}^{2 - k} \cdot {(- \frac{1}{3})}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} {(x)}^{2 - 0} \cdot {(- \frac{1}{3})}^{0} + \frac{2!}{(2 - 1)! 1!} {(x)}^{2 - 1} \cdot {(- \frac{1}{3})}^{1} + \frac{2!}{(2 - 2)! 2!} {(x)}^{2 - 2} \cdot {(- \frac{1}{3})}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{2} \cdot {(- \frac{1}{3})}^{0} + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(x)}^{2}$ by $1$ .

${(x)}^{2} \cdot {(- \frac{1}{3})}^{0} + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.2

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

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Step 4.2.1

Apply the product rule to $- \frac{1}{3}$ .

$x^{2} \cdot ({(- 1)}^{0} {(\frac{1}{3})}^{0}) + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.2.2

Apply the product rule to $\frac{1}{3}$ .

$x^{2} \cdot ({(- 1)}^{0} \frac{1^{0}}{3^{0}}) + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.3

Anything raised to $0$ is $1$ .

$x^{2} \cdot (1 \frac{1^{0}}{3^{0}}) + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.4

Multiply $\frac{1^{0}}{3^{0}}$ by $1$ .

$x^{2} \cdot \frac{1^{0}}{3^{0}} + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.5

Anything raised to $0$ is $1$ .

$x^{2} \cdot \frac{1}{3^{0}} + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.6

Anything raised to $0$ is $1$ .

$x^{2} \cdot \frac{1}{1} + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.7

Divide $1$ by $1$ .

$x^{2} \cdot 1 + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.8

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

$x^{2} + 2 \cdot {(x)}^{1} \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.9

Simplify.

$x^{2} + 2 \cdot x \cdot {(- \frac{1}{3})}^{1} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.10

Simplify.

$x^{2} + 2 x \cdot (- \frac{1}{3}) + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.11

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

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Step 4.11.1

Multiply $- 1$ by $2$ .

$x^{2} - 2 x \frac{1}{3} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.11.2

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

$x^{2} + x \frac{- 2}{3} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.11.3

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

$x^{2} + \frac{x \cdot - 2}{3} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.12

Move $- 2$ to the left of $x$ .

$x^{2} + \frac{- 2 \cdot x}{3} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.13

Move the negative in front of the fraction.

$x^{2} - \frac{2 \cdot x}{3} + 1 \cdot {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.14

Multiply ${(x)}^{0}$ by $1$ .

$x^{2} - \frac{2 x}{3} + {(x)}^{0} \cdot {(- \frac{1}{3})}^{2}$

Step 4.15

Anything raised to $0$ is $1$ .

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

Step 4.16

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

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

Step 4.17

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

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Step 4.17.1

Apply the product rule to $- \frac{1}{3}$ .

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

Step 4.17.2

Apply the product rule to $\frac{1}{3}$ .

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

Step 4.18

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

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

Step 4.19

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

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

Step 4.20

One to any power is one.

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

Step 4.21

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

$x^{2} - \frac{2 x}{3} + \frac{1}{9}$