Algebra Examples

{(2 x + 1)}^{2}

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

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

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

Step 2

Expand the summation.

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

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

Step 3

Simplify the exponents for each term of the expansion.

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

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

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply

1

$1$ by

{(1)}^{0}

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

Tap for more steps...

Step 4.1.1

Move

{(1)}^{0}

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

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

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

Step 4.1.2

Multiply

{(1)}^{0}

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

1

$1$ .

Tap for more steps...

Step 4.1.2.1

Raise

1

$1$ to the power of

1

$1$ .

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

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

Step 4.1.2.2

Use the power rule

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

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

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

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

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

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

Step 4.1.3

Add

0

$0$ and

1

$1$ .

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

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

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

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

Step 4.2

Simplify

1^{1} \cdot {(2 x)}^{2}

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

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

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

Step 4.3

Apply the product rule to

2 x

$2 x$ .

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

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

Step 4.4

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 4.5

Simplify.

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

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

Step 4.6

Multiply

2

$2$ by

2

$2$ .

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

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

Step 4.7

Evaluate the exponent.

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

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

Step 4.8

Multiply

4

$4$ by

1

$1$ .

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

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

Step 4.9

Multiply

1

$1$ by

{(1)}^{2}

${(1)}^{2}$ by adding the exponents.

Tap for more steps...

Step 4.9.1

Move

{(1)}^{2}

${(1)}^{2}$ .

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

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

Step 4.9.2

Multiply

{(1)}^{2}

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

1

$1$ .

Tap for more steps...

Step 4.9.2.1

Raise

1

$1$ to the power of

1

$1$ .

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

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

Step 4.9.2.2

Use the power rule

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

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

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

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

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

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

Step 4.9.3

Add

2

$2$ and

1

$1$ .

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

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

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

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

Step 4.10

Simplify

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

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

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

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

Step 4.11

One to any power is one.

4 x^{2} + 4 x + 1

$4 x^{2} + 4 x + 1$

4 x^{2} + 4 x + 1

$4 x^{2} + 4 x + 1$