expand (x+4)^3 — step-by-step solution

Let's write in expanded form:

{(x + 4)}^{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

{(x + 4)}^{3}

\frac{3!}{3! \cdot 0!} \cdot x^{3} \cdot 4^{0}

+ \frac{3!}{2! \cdot 1!} \cdot x^{2} \cdot 4^{1}

+ \frac{3!}{1! \cdot 2!} \cdot x^{1} \cdot 4^{2}

+ \frac{3!}{0! \cdot 3!} \cdot x^{0} \cdot 4^{3}

\frac{3!}{3! \cdot 0!} \cdot x^{3} \cdot 4^{0}

+ \frac{3!}{2! \cdot 1!} \cdot x^{2} \cdot 4^{1}

+ \frac{3!}{1! \cdot 2!} \cdot x^{1} \cdot 4^{2}

+ \frac{3!}{0! \cdot 3!} \cdot x^{0} \cdot 4^{3}

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

4^{0}

1

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

3!

6

1!

1

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

2!

2

1!

1

■Multiplication by 1 preserves the original value:

2 \cdot 1

2

■Multiplying by 1 changes nothing:

1 \cdot 2

2

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

4^{2}

4 \cdot 4

16

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

0^{0} = 1

x^{0}

1

\frac{3!}{3! \cdot 0!} \cdot x^{3}

+ \frac{6}{2} \cdot x^{2} \cdot 4

+ \frac{6}{2} \cdot x \cdot 16

+ \frac{3!}{0! \cdot 3!} \cdot 4^{3}

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

3!

6

■The factorial of 0 is defined to be 1:

0!

1

■Multiplication by 1 preserves the original value:

6 \cdot 1

6

■Perform multiplications before additions and subtractions:

\frac{6}{2} \cdot 4

6 \cdot 2

\frac{6}{2} \cdot 16

6 \cdot 8

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

3!

6

■By convention, the factorial of 0 is 1:

0!

1

■Multiplying by 1 changes nothing:

1 \cdot 6

6

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

4^{3}

4 \cdot 4 \cdot 4

64

\frac{6}{6} \cdot x^{3}

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

+ 6 \cdot 8 \cdot x

+ \frac{6}{6} \cdot 64

ⓘ 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 x^{3}

x^{3}

1 \cdot 64

64

■Multiplications are done before additions and subtractions:

6 \cdot 2

12

6 \cdot 8

48

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

6

, their greatest common divisor:

\frac{6}{6}

1

▶Expanded form:

x^{3}

+ 12 x^{2}

+ 48 x

+ 64