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

Let's write in expanded form:

{(x + 3)}^{3}

Using the Newton formula:

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

, you can expand the math expression:

ⅰ

{(x + 3)}^{3}

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

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

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

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

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

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

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

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

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

3^{0}

1

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

3!

6

2!

2

1!

1

■Multiplying by 1 changes nothing:

2 \cdot 1

2

1 \cdot 2

2

■Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

3^{2}

3 \cdot 3

9

■Any nonzero number raised to the power 0 is 1. In this calculation, we use the convention 0^0 = 1:

x^{0}

1

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

+ \frac{6}{2} \cdot x^{2} \cdot 3

+ \frac{6}{2} \cdot x \cdot 9

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

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

3!

6

■By convention, the factorial of 0 is 1:

0!

1

■Multiplying by 1 changes nothing:

6 \cdot 1

6

1 \cdot 6

6

■Multiplications are done before additions and subtractions:

\frac{6}{2} \cdot 3

\frac{18}{2}

\frac{6}{2} \cdot 9

\frac{54}{2}

■Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

3^{3}

3 \cdot 3 \cdot 3

27

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

+ \frac{18}{2} \cdot x^{2}

+ \frac{54}{2} \cdot x

+ \frac{6}{6} \cdot 27

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

6

, their greatest common divisor:

\frac{6}{6}

1

■Multiplying by 1 changes nothing:

1 \cdot x^{3}

x^{3}

1 \cdot 27

27

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

2

, their greatest common divisor:

\frac{18}{2}

9

\frac{54}{2}

27

▶Expanded form:

x^{3}

+ 9 x^{2}

+ 27 x

+ 27