Algebra Examples

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

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

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})

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

2 \sum k = 0 \frac{2!}{(2 - k)! k!} \cdot {(3 x)}^{2 - k} \cdot {(- 1)}^{k}

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

Step 2

Expand the summation.

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

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

Step 3

Simplify the exponents for each term of the expansion.

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

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

Step 4

Simplify each term.

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Step 4.1

Multiply

{(3 x)}^{2}

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

1

$1$ .

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

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

Step 4.2

Apply the product rule to

3 x

$3 x$ .

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

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

Step 4.3

Raise

3

$3$ to the power of

2

$2$ .

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

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

Step 4.4

Anything raised to

0

$0$ is

1

$1$ .

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

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

Step 4.5

Multiply

9

$9$ by

1

$1$ .

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

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

Step 4.6

Simplify.

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

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

Step 4.7

Multiply

3

$3$ by

2

$2$ .

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

$9 x^{2} + 6 x \cdot {(- 1)}^{1} + 1 \cdot {(3 x)}^{0} \cdot {(- 1)}^{2}$

Step 4.8

Evaluate the exponent.

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

$9 x^{2} + 6 x \cdot - 1 + 1 \cdot {(3 x)}^{0} \cdot {(- 1)}^{2}$

Step 4.9

Multiply

- 1

$- 1$ by

6

$6$ .

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

$9 x^{2} - 6 x + 1 \cdot {(3 x)}^{0} \cdot {(- 1)}^{2}$

Step 4.10

Multiply

{(3 x)}^{0}

${(3 x)}^{0}$ by

1

$1$ .

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

$9 x^{2} - 6 x + {(3 x)}^{0} \cdot {(- 1)}^{2}$

Step 4.11

Apply the product rule to

3 x

$3 x$ .

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

$9 x^{2} - 6 x + 3^{0} x^{0} \cdot {(- 1)}^{2}$

Step 4.12

Anything raised to

0

$0$ is

1

$1$ .

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

$9 x^{2} - 6 x + 1 x^{0} \cdot {(- 1)}^{2}$

Step 4.13

Multiply

x^{0}

$x^{0}$ by

1

$1$ .

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

$9 x^{2} - 6 x + x^{0} \cdot {(- 1)}^{2}$

Step 4.14

Anything raised to

0

$0$ is

1

$1$ .

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

$9 x^{2} - 6 x + 1 \cdot {(- 1)}^{2}$

Step 4.15

Multiply

{(- 1)}^{2}

${(- 1)}^{2}$ by

1

$1$ .

9 x^{2} - 6 x + {(- 1)}^{2}

$9 x^{2} - 6 x + {(- 1)}^{2}$

Step 4.16

Raise

- 1

$- 1$ to the power of

2

$2$ .

9 x^{2} - 6 x + 1

$9 x^{2} - 6 x + 1$

9 x^{2} - 6 x + 1

$9 x^{2} - 6 x + 1$