Algebra Examples

{(x - 1)}^{4}

${(x - 1)}^{4}$

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})

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

4 \sum k = 0 \frac{4!}{(4 - k)! k!} \cdot {(x)}^{4 - k} \cdot {(- 1)}^{k}

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(x)}^{4 - k} \cdot {(- 1)}^{k}$

Step 2

Expand the summation.

\frac{4!}{(4 - 0)! 0!} {(x)}^{4 - 0} \cdot {(- 1)}^{0} + \frac{4!}{(4 - 1)! 1!} {(x)}^{4 - 1} \cdot {(- 1)}^{1} + \frac{4!}{(4 - 2)! 2!} {(x)}^{4 - 2} \cdot {(- 1)}^{2} + \frac{4!}{(4 - 3)! 3!} {(x)}^{4 - 3} \cdot {(- 1)}^{3} + \frac{4!}{(4 - 4)! 4!} {(x)}^{4 - 4} \cdot {(- 1)}^{4}

$\frac{4!}{(4 - 0)! 0!} {(x)}^{4 - 0} \cdot {(- 1)}^{0} + \frac{4!}{(4 - 1)! 1!} {(x)}^{4 - 1} \cdot {(- 1)}^{1} + \frac{4!}{(4 - 2)! 2!} {(x)}^{4 - 2} \cdot {(- 1)}^{2} + \frac{4!}{(4 - 3)! 3!} {(x)}^{4 - 3} \cdot {(- 1)}^{3} + \frac{4!}{(4 - 4)! 4!} {(x)}^{4 - 4} \cdot {(- 1)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

1 \cdot {(x)}^{4} \cdot {(- 1)}^{0} + 4 \cdot {(x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$1 \cdot {(x)}^{4} \cdot {(- 1)}^{0} + 4 \cdot {(x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply

{(x)}^{4}

${(x)}^{4}$ by

1

$1$ .

{(x)}^{4} \cdot {(- 1)}^{0} + 4 \cdot {(x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

${(x)}^{4} \cdot {(- 1)}^{0} + 4 \cdot {(x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.2

Anything raised to

0

$0$ is

1

$1$ .

x^{4} \cdot 1 + 4 \cdot {(x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} \cdot 1 + 4 \cdot {(x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.3

Multiply

x^{4}

$x^{4}$ by

1

$1$ .

x^{4} + 4 \cdot {(x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} + 4 \cdot {(x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.4

Evaluate the exponent.

x^{4} + 4 x^{3} \cdot - 1 + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} + 4 x^{3} \cdot - 1 + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.5

Multiply

- 1

$- 1$ by

4

$4$ .

x^{4} - 4 x^{3} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 \cdot {(x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.6

Raise

- 1

$- 1$ to the power of

2

$2$ .

x^{4} - 4 x^{3} + 6 x^{2} \cdot 1 + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 x^{2} \cdot 1 + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.7

Multiply

6

$6$ by

1

$1$ .

x^{4} - 4 x^{3} + 6 x^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 x^{2} + 4 \cdot {(x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.8

Simplify.

x^{4} - 4 x^{3} + 6 x^{2} + 4 \cdot x \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 x^{2} + 4 \cdot x \cdot {(- 1)}^{3} + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.9

Raise

- 1

$- 1$ to the power of

3

$3$ .

x^{4} - 4 x^{3} + 6 x^{2} + 4 x \cdot - 1 + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 x^{2} + 4 x \cdot - 1 + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.10

Multiply

- 1

$- 1$ by

4

$4$ .

x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1 \cdot {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.11

Multiply

{(x)}^{0}

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

1

$1$ .

x^{4} - 4 x^{3} + 6 x^{2} - 4 x + {(x)}^{0} \cdot {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 x^{2} - 4 x + {(x)}^{0} \cdot {(- 1)}^{4}$

Step 4.12

Anything raised to

0

$0$ is

1

$1$ .

x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1 \cdot {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1 \cdot {(- 1)}^{4}$

Step 4.13

Multiply

{(- 1)}^{4}

${(- 1)}^{4}$ by

1

$1$ .

x^{4} - 4 x^{3} + 6 x^{2} - 4 x + {(- 1)}^{4}

$x^{4} - 4 x^{3} + 6 x^{2} - 4 x + {(- 1)}^{4}$

Step 4.14

Raise

- 1

$- 1$ to the power of

4

$4$ .

x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1

$x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$

x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1

$x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$