Algebra Examples

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

${(2 x - 1)}^{4}$

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(2 x)}^{4 - k} \cdot {(- 1)}^{k}$

Step 2

Expand the summation.

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

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

Step 3

Simplify the exponents for each term of the expansion.

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

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

Step 4

Simplify each term.

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Step 4.1

Multiply

{(2 x)}^{4}

${(2 x)}^{4}$ by

1

$1$ .

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

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

Step 4.2

Apply the product rule to

2 x

$2 x$ .

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

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

Step 4.3

Raise

2

$2$ to the power of

4

$4$ .

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

$16 x^{4} \cdot {(- 1)}^{0} + 4 \cdot {(2 x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.4

Anything raised to

0

$0$ is

1

$1$ .

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

$16 x^{4} \cdot 1 + 4 \cdot {(2 x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.5

Multiply

16

$16$ by

1

$1$ .

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

$16 x^{4} + 4 \cdot {(2 x)}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.6

Apply the product rule to

2 x

$2 x$ .

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

$16 x^{4} + 4 \cdot (2^{3} x^{3}) \cdot {(- 1)}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.7

Raise

2

$2$ to the power of

3

$3$ .

16 x^{4} + 4 \cdot (8 x^{3}) \cdot {(- 1)}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} + 4 \cdot (8 x^{3}) \cdot {(- 1)}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.8

Multiply

8

$8$ by

4

$4$ .

16 x^{4} + 32 x^{3} \cdot {(- 1)}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} + 32 x^{3} \cdot {(- 1)}^{1} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.9

Evaluate the exponent.

16 x^{4} + 32 x^{3} \cdot - 1 + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} + 32 x^{3} \cdot - 1 + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.10

Multiply

- 1

$- 1$ by

32

$32$ .

16 x^{4} - 32 x^{3} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 6 \cdot {(2 x)}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.11

Apply the product rule to

2 x

$2 x$ .

16 x^{4} - 32 x^{3} + 6 \cdot (2^{2} x^{2}) \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 6 \cdot (2^{2} x^{2}) \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.12

Raise

2

$2$ to the power of

2

$2$ .

16 x^{4} - 32 x^{3} + 6 \cdot (4 x^{2}) \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 6 \cdot (4 x^{2}) \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.13

Multiply

4

$4$ by

6

$6$ .

16 x^{4} - 32 x^{3} + 24 x^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} \cdot {(- 1)}^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.14

Raise

- 1

$- 1$ to the power of

2

$2$ .

16 x^{4} - 32 x^{3} + 24 x^{2} \cdot 1 + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} \cdot 1 + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.15

Multiply

24

$24$ by

1

$1$ .

16 x^{4} - 32 x^{3} + 24 x^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} + 4 \cdot {(2 x)}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.16

Simplify.

16 x^{4} - 32 x^{3} + 24 x^{2} + 4 \cdot (2 x) \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} + 4 \cdot (2 x) \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.17

Multiply

2

$2$ by

4

$4$ .

16 x^{4} - 32 x^{3} + 24 x^{2} + 8 x \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} + 8 x \cdot {(- 1)}^{3} + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.18

Raise

- 1

$- 1$ to the power of

3

$3$ .

16 x^{4} - 32 x^{3} + 24 x^{2} + 8 x \cdot - 1 + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} + 8 x \cdot - 1 + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.19

Multiply

- 1

$- 1$ by

8

$8$ .

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1 \cdot {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.20

Multiply

{(2 x)}^{0}

${(2 x)}^{0}$ by

1

$1$ .

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + {(2 x)}^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + {(2 x)}^{0} \cdot {(- 1)}^{4}$

Step 4.21

Apply the product rule to

2 x

$2 x$ .

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 2^{0} x^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 2^{0} x^{0} \cdot {(- 1)}^{4}$

Step 4.22

Anything raised to

0

$0$ is

1

$1$ .

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1 x^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1 x^{0} \cdot {(- 1)}^{4}$

Step 4.23

Multiply

x^{0}

$x^{0}$ by

1

$1$ .

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + x^{0} \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + x^{0} \cdot {(- 1)}^{4}$

Step 4.24

Anything raised to

0

$0$ is

1

$1$ .

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1 \cdot {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1 \cdot {(- 1)}^{4}$

Step 4.25

Multiply

{(- 1)}^{4}

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

1

$1$ .

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + {(- 1)}^{4}

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + {(- 1)}^{4}$

Step 4.26

Raise

- 1

$- 1$ to the power of

4

$4$ .

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1$

16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1

$16 x^{4} - 32 x^{3} + 24 x^{2} - 8 x + 1$