Examples

{(x - 7)}^{3}

${(x - 7)}^{3}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states

{(a + b)}^{n} = n \sum k = 0 n C k \cdot (a^{n - k} b^{k})

${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

3 \sum k = 0 \frac{3!}{(3 - k)! k!} \cdot {(x)}^{3 - k} \cdot {(- 7)}^{k}

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(x)}^{3 - k} \cdot {(- 7)}^{k}$

Step 2

Expand the summation.

\frac{3!}{(3 - 0)! 0!} {(x)}^{3 - 0} \cdot {(- 7)}^{0} + \frac{3!}{(3 - 1)! 1!} {(x)}^{3 - 1} \cdot {(- 7)}^{1} + \frac{3!}{(3 - 2)! 2!} {(x)}^{3 - 2} \cdot {(- 7)}^{2} + \frac{3!}{(3 - 3)! 3!} {(x)}^{3 - 3} \cdot {(- 7)}^{3}

$\frac{3!}{(3 - 0)! 0!} {(x)}^{3 - 0} \cdot {(- 7)}^{0} + \frac{3!}{(3 - 1)! 1!} {(x)}^{3 - 1} \cdot {(- 7)}^{1} + \frac{3!}{(3 - 2)! 2!} {(x)}^{3 - 2} \cdot {(- 7)}^{2} + \frac{3!}{(3 - 3)! 3!} {(x)}^{3 - 3} \cdot {(- 7)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

1 \cdot {(x)}^{3} \cdot {(- 7)}^{0} + 3 \cdot {(x)}^{2} \cdot {(- 7)}^{1} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}

$1 \cdot {(x)}^{3} \cdot {(- 7)}^{0} + 3 \cdot {(x)}^{2} \cdot {(- 7)}^{1} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4

Simplify each term.

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Step 4.1

Multiply

{(x)}^{3}

${(x)}^{3}$ by

1

$1$ .

{(x)}^{3} \cdot {(- 7)}^{0} + 3 \cdot {(x)}^{2} \cdot {(- 7)}^{1} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}

${(x)}^{3} \cdot {(- 7)}^{0} + 3 \cdot {(x)}^{2} \cdot {(- 7)}^{1} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.2

Anything raised to

0

$0$ is

1

$1$ .

x^{3} \cdot 1 + 3 \cdot {(x)}^{2} \cdot {(- 7)}^{1} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}

$x^{3} \cdot 1 + 3 \cdot {(x)}^{2} \cdot {(- 7)}^{1} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.3

Multiply

x^{3}

$x^{3}$ by

1

$1$ .

x^{3} + 3 \cdot {(x)}^{2} \cdot {(- 7)}^{1} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}

$x^{3} + 3 \cdot {(x)}^{2} \cdot {(- 7)}^{1} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.4

Evaluate the exponent.

x^{3} + 3 x^{2} \cdot - 7 + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}

$x^{3} + 3 x^{2} \cdot - 7 + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.5

Multiply

$- 7$ by

$3$ .

$x^{3} - 21 x^{2} + 3 \cdot {(x)}^{1} \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.6

Simplify.

$x^{3} - 21 x^{2} + 3 \cdot x \cdot {(- 7)}^{2} + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.7

Raise

$- 7$ to the power of

$2$ .

$x^{3} - 21 x^{2} + 3 x \cdot 49 + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.8

Multiply

$49$ by

$3$ .

$x^{3} - 21 x^{2} + 147 x + 1 \cdot {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.9

Multiply

${(x)}^{0}$ by

$1$ .

$x^{3} - 21 x^{2} + 147 x + {(x)}^{0} \cdot {(- 7)}^{3}$

Step 4.10

Anything raised to

$0$ is

$1$ .

$x^{3} - 21 x^{2} + 147 x + 1 \cdot {(- 7)}^{3}$

Step 4.11

Multiply

${(- 7)}^{3}$ by

$1$ .

$x^{3} - 21 x^{2} + 147 x + {(- 7)}^{3}$

Step 4.12

Raise

$- 7$ to the power of

$3$ .

$x^{3} - 21 x^{2} + 147 x - 343$

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