Algebra Examples

{(1 - x)}^{4}

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

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})

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

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

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

Step 2

Expand the summation.

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

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

Step 3

Simplify the exponents for each term of the expansion.

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

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

Step 4

Simplify each term.

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Step 4.1

Multiply

1

$1$ by

{(1)}^{4}

${(1)}^{4}$ by adding the exponents.

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Step 4.1.1

Multiply

1

$1$ by

{(1)}^{4}

${(1)}^{4}$ .

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Step 4.1.1.1

Raise

1

$1$ to the power of

1

$1$ .

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

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

Step 4.1.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.1.2

Add

1

$1$ and

4

$4$ .

1^{5} \cdot {(- x)}^{0} + 4 \cdot {(1)}^{3} \cdot {(- x)}^{1} + 6 \cdot {(1)}^{2} \cdot {(- x)}^{2} + 4 \cdot {(1)}^{1} \cdot {(- x)}^{3} + 1 \cdot {(1)}^{0} \cdot {(- x)}^{4}

$1^{5} \cdot {(- x)}^{0} + 4 \cdot {(1)}^{3} \cdot {(- x)}^{1} + 6 \cdot {(1)}^{2} \cdot {(- x)}^{2} + 4 \cdot {(1)}^{1} \cdot {(- x)}^{3} + 1 \cdot {(1)}^{0} \cdot {(- x)}^{4}$

1^{5} \cdot {(- x)}^{0} + 4 \cdot {(1)}^{3} \cdot {(- x)}^{1} + 6 \cdot {(1)}^{2} \cdot {(- x)}^{2} + 4 \cdot {(1)}^{1} \cdot {(- x)}^{3} + 1 \cdot {(1)}^{0} \cdot {(- x)}^{4}

$1^{5} \cdot {(- x)}^{0} + 4 \cdot {(1)}^{3} \cdot {(- x)}^{1} + 6 \cdot {(1)}^{2} \cdot {(- x)}^{2} + 4 \cdot {(1)}^{1} \cdot {(- x)}^{3} + 1 \cdot {(1)}^{0} \cdot {(- x)}^{4}$

Step 4.2

Simplify

1^{5} \cdot {(- x)}^{0}

$1^{5} \cdot {(- x)}^{0}$ .

1^{5} + 4 \cdot {(1)}^{3} \cdot {(- x)}^{1} + 6 \cdot {(1)}^{2} \cdot {(- x)}^{2} + 4 \cdot {(1)}^{1} \cdot {(- x)}^{3} + 1 \cdot {(1)}^{0} \cdot {(- x)}^{4}

$1^{5} + 4 \cdot {(1)}^{3} \cdot {(- x)}^{1} + 6 \cdot {(1)}^{2} \cdot {(- x)}^{2} + 4 \cdot {(1)}^{1} \cdot {(- x)}^{3} + 1 \cdot {(1)}^{0} \cdot {(- x)}^{4}$

Step 4.3

One to any power is one.

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

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

Step 4.4

One to any power is one.

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

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

Step 4.5

Multiply

4

$4$ by

1

$1$ .

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

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

Step 4.6

Simplify.

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

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

Step 4.7

Multiply

- 1

$- 1$ by

4

$4$ .

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

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

Step 4.8

One to any power is one.

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

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

Step 4.9

Multiply

6

$6$ by

1

$1$ .

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

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

Step 4.10

Apply the product rule to

- x

$- x$ .

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

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

Step 4.11

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 4.12

Multiply

x^{2}

$x^{2}$ by

1

$1$ .

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

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

Step 4.13

Evaluate the exponent.

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

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

Step 4.14

Multiply

4

$4$ by

1

$1$ .

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

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

Step 4.15

Apply the product rule to

- x

$- x$ .

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

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

Step 4.16

Raise

- 1

$- 1$ to the power of

3

$3$ .

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

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

Step 4.17

Multiply

- 1

$- 1$ by

4

$4$ .

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

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

Step 4.18

Multiply

1

$1$ by

{(1)}^{0}

${(1)}^{0}$ by adding the exponents.

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Step 4.18.1

Multiply

1

$1$ by

{(1)}^{0}

${(1)}^{0}$ .

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Step 4.18.1.1

Raise

1

$1$ to the power of

1

$1$ .

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

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

Step 4.18.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.18.2

Add

1

$1$ and

0

$0$ .

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

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

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

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

Step 4.19

Simplify

1^{1} \cdot {(- x)}^{4}

$1^{1} \cdot {(- x)}^{4}$ .

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

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

Step 4.20

Apply the product rule to

- x

$- x$ .

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

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

Step 4.21

Raise

- 1

$- 1$ to the power of

4

$4$ .

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

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

Step 4.22

Multiply

x^{4}

$x^{4}$ by

1

$1$ .

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

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

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

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