Algebra Examples

${(1 - x)}^{3}$

Step 1

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

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

Multiply

$1$ by

${(1)}^{3}$ by adding the exponents.

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Step 4.1.1

Multiply

$1$ by

${(1)}^{3}$ .

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Step 4.1.1.1

Raise

$1$ to the power of

$1$ .

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

Step 4.1.1.2

Use the power rule

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

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

Step 4.1.2

Add

$1$ and

$3$ .

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

Step 4.2

Simplify

$1^{4} \cdot {(- x)}^{0}$ .

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

Step 4.3

One to any power is one.

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

Step 4.4

One to any power is one.

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

Step 4.5

Multiply

$3$ by

$1$ .

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

Step 4.6

Simplify.

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

Step 4.7

Multiply

$- 1$ by

$3$ .

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

Step 4.8

Evaluate the exponent.

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

Step 4.9

Multiply

$3$ by

$1$ .

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

Step 4.10

Apply the product rule to

$- x$ .

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

Step 4.11

Raise

$- 1$ to the power of

$2$ .

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

Step 4.12

Multiply

$x^{2}$ by

$1$ .

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

Step 4.13

Multiply

$1$ by

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

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Step 4.13.1

Multiply

$1$ by

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

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Step 4.13.1.1

Raise

$1$ to the power of

$1$ .

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

Step 4.13.1.2

Use the power rule

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

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

Step 4.13.2

Add

$1$ and

$0$ .

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

Step 4.14

Simplify

$1^{1} \cdot {(- x)}^{3}$ .

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

Step 4.15

Apply the product rule to

$- x$ .

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

Step 4.16

Raise

$- 1$ to the power of

$3$ .

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