expand (2x+1)^3 — step-by-step solution

Let's expand:

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

Using the Newton formula:

{(a + b)}^{n} = \sum_{k = 0}^{n} (\frac{n!}{k! (n - k)!}) a^{n - k} b^{k}

, you can expand the math expression:

ⅰ

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

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

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

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

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

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

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

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

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

ⓘ Any nonzero number raised to the power 0 is 1:

1^{0}

1

■Raising a nonzero number to the power 0 gives 1. The convention used for this calculation is

0^{0} = 1

{(2 x)}^{0}

1

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

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

+ \frac{3!}{1! \cdot 2!} \cdot 2 x

+ \frac{3!}{0! \cdot 3!}

ⓘ The factorial of a positive integer is the product of all the integers from 1 to that integer:

3!

6

2!

2

1!

1

■Calculate the factorial of a positive integer by multiplying all integers from 1 to that integer:

3!

6

1!

1

■The factorial of 0 is defined to be 1:

0!

1

■Multiplication by 1 preserves the original value:

6 \cdot 1

6

■Multiplying by 1 changes nothing:

2 \cdot 1

2

1 \cdot 2

2

1 \cdot 6

6

■By convention, the factorial of 0 is 1:

0!

1

\frac{6}{6} \cdot {(2 x)}^{3}

+ \frac{6}{2} \cdot {(2 x)}^{2}

+ \frac{6}{2} \cdot 2 x

+ \frac{6}{6}

ⓘ Simplify the fraction by dividing the numerator and the denominator by

6

, their greatest common divisor:

\frac{6}{6}

1

■Multiplication by 1 preserves the original value:

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

{(2 x)}^{3}

■We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{6}{2}

3

■Multiplications are done before additions and subtractions:

\frac{6}{2} \cdot 2

6

■We simplify the fraction by dividing the numerator and the denominator by

6

, their greatest common divisor:

\frac{6}{6}

1

{(2 x)}^{3}

+ 3 \cdot {(2 x)}^{2}

+ 6 x

+ 1

ⓘ The power of a product is the product of the powers:

{(2 x)}^{3}

2^{3} \cdot x^{3}

{(2 x)}^{2}

2^{2} \cdot x^{2}

2^{3} \cdot x^{3}

+ 3 \cdot 2^{2} \cdot x^{2}

+ 6 x

+ 1

ⓘ Calculate powers before multiplications, additions and subtractions. A power with a positive integer exponent is a repeated product:

2^{3}

2 \cdot 2 \cdot 2

8

■Calculate powers before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product:

2^{2}

2 \cdot 2

4

8 x^{3}

+ 3 \cdot 4 x^{2}

+ 6 x

+ 1

ⓘ Multiplications are done before additions and subtractions:

3 \cdot 4

12

▶Expanded form:

8 x^{3}

+ 12 x^{2}

+ 6 x

+ 1