# How do you use the binomial series to expand (5 + x)^4?

Apr 17, 2018

${\left(5 + x\right)}^{4} = 625 + 500 x + 150 {x}^{2} + 20 {x}^{3} + {x}^{4}$

#### Explanation:

The binomial series expansion for (a+bx)^n,ninZZ;n>0 is given by:
(a+bx)^n=sum_(r=0)^n((n!)/(r!(n-1)!)a^(n-r)(bx)^r)

So, we have:
(5+x)^4=(4!)/(0!*4!)5^4+(4!)/(1!*3!)(5)^3x+(4!)/(2!*2!)(5)^2x^2+(4!)/(4!*1!)(5)x^3+(4!)/(4!*0!)x^4

${\left(5 + x\right)}^{4} = {5}^{4} + 4 {\left(5\right)}^{3} x + 6 {\left(5\right)}^{2} {x}^{2} + 4 \left(5\right) {x}^{3} + {x}^{4}$

${\left(5 + x\right)}^{4} = 625 + 500 x + 150 {x}^{2} + 20 {x}^{3} + {x}^{4}$