Question

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

Answer 1

`(5+x)^4=625+500x+150x^2+20x^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`

`(5+x)^4=5^4+4(5)^3x+6(5)^2x^2+4(5)x^3+x^4`

`(5+x)^4=625+500x+150x^2+20x^3+x^4`