Question

How do you use the binomial formula to expand ${(x+1)}^3$

Answer 1

Expand ${(x+1)}^3$

Binomial formula to expand ${\left(a+b\right)}^3$ is

${\left(a+b\right)}^3={}^{3}C_0{a}^3{b}^0+{}^{3}C_1{a}^2{b}^1+{}^{3}C_2{a}^1{b}^2+{}^{3}C_3{a}^0{b}^3$

Here we have, $a=x$ and $b=1$

Substituting the values of $a$ and $b$ we get

$\Rightarrow$${\left(x+1\right)}^3={}^{3}C_0{x}^3{1}^0+{}^{3}C_1{x}^2{1}^1+{}^{3}C_2{x}^1{1}^2+{}^{3}C_3{x}^0{1}^3$

$\Rightarrow$${\left(x+1\right)}^3=\frac{3!}{0!3!}{x}^3{1}^0+\frac{3!}{1!2!}{x}^2{1}^1+\frac{3!}{2!1!}{x}^1{1}^2+\frac{3!}{3!0!}{x}^0{1}^3$ $...\left[\because {}^n{C}_r=\frac{n!}{r!\left(n-r\right)!}\right]$

$\Rightarrow$${\left(x+1\right)}^3=x^3+3x^2+3x+1$

Hence the expansion of ${(x+1)}^3$ using binomial theorem is $x^3+3x^2+3x+1$