Algebra Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(3 x)}^{3 - k} \cdot {(- 2)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

Multiply ${(3 x)}^{3}$ by $1$ .

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

Step 4.2

Apply the product rule to $3 x$ .

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

Step 4.3

Raise $3$ to the power of $3$ .

$27 x^{3} \cdot {(- 2)}^{0} + 3 \cdot {(3 x)}^{2} \cdot {(- 2)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.4

Anything raised to $0$ is $1$ .

$27 x^{3} \cdot 1 + 3 \cdot {(3 x)}^{2} \cdot {(- 2)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.5

Multiply $27$ by $1$ .

$27 x^{3} + 3 \cdot {(3 x)}^{2} \cdot {(- 2)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.6

Apply the product rule to $3 x$ .

$27 x^{3} + 3 \cdot (3^{2} x^{2}) \cdot {(- 2)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.7

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 4.7.1

Move $3^{2}$ .

$27 x^{3} + 3^{2} \cdot 3 \cdot x^{2} \cdot {(- 2)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.7.2

Multiply $3^{2}$ by $3$ .

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Step 4.7.2.1

Raise $3$ to the power of $1$ .

$27 x^{3} + 3^{2} \cdot 3^{1} \cdot x^{2} \cdot {(- 2)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 x^{3} + 3^{2 + 1} \cdot x^{2} \cdot {(- 2)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.7.3

Add $2$ and $1$ .

$27 x^{3} + 3^{3} \cdot x^{2} \cdot {(- 2)}^{1} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.8

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

$27 x^{3} + 3^{3} \cdot x^{2} \cdot - 2 + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.9

Raise $3$ to the power of $3$ .

$27 x^{3} + 27 \cdot x^{2} \cdot - 2 + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.10

Multiply $- 2$ by $27$ .

$27 x^{3} - 54 x^{2} + 3 \cdot {(3 x)}^{1} \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.11

Simplify.

$27 x^{3} - 54 x^{2} + 3 \cdot (3 x) \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.12

Multiply $3$ by $3$ .

$27 x^{3} - 54 x^{2} + 9 x \cdot {(- 2)}^{2} + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.13

Raise $- 2$ to the power of $2$ .

$27 x^{3} - 54 x^{2} + 9 x \cdot 4 + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.14

Multiply $4$ by $9$ .

$27 x^{3} - 54 x^{2} + 36 x + 1 \cdot {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.15

Multiply ${(3 x)}^{0}$ by $1$ .

$27 x^{3} - 54 x^{2} + 36 x + {(3 x)}^{0} \cdot {(- 2)}^{3}$

Step 4.16

Apply the product rule to $3 x$ .

$27 x^{3} - 54 x^{2} + 36 x + 3^{0} x^{0} \cdot {(- 2)}^{3}$

Step 4.17

Anything raised to $0$ is $1$ .

$27 x^{3} - 54 x^{2} + 36 x + 1 x^{0} \cdot {(- 2)}^{3}$

Step 4.18

Multiply $x^{0}$ by $1$ .

$27 x^{3} - 54 x^{2} + 36 x + x^{0} \cdot {(- 2)}^{3}$

Step 4.19

Anything raised to $0$ is $1$ .

$27 x^{3} - 54 x^{2} + 36 x + 1 \cdot {(- 2)}^{3}$

Step 4.20

Multiply ${(- 2)}^{3}$ by $1$ .

$27 x^{3} - 54 x^{2} + 36 x + {(- 2)}^{3}$

Step 4.21

Raise $- 2$ to the power of $3$ .

$27 x^{3} - 54 x^{2} + 36 x - 8$