How do you use the binomial series to expand (1+x^2)^5?

1 Answer
Mar 7, 2018

1+5x^2+10x^4+10x^6+5x^8+x^10

Explanation:

For a binomial expansion:

(x+y)^n we have:

((n),(r))x^(n-r)y^r

sum_(r=0)^(n)((n),(r))x^(n-r)y^r

Where:

((n),(r))=color(white)(0)^n C_(r)=(n!)/(r!(n-r)!)

Beginning with r=0

((5),(0))1^5(x^2)^0+((5),(1))1^4(x^2)^1+((5),(2))1^3(x^2)^2+((5),(3))1^2(x^2)^3

->((5),(4))1^1(x^2)^4+((5),(5))1^0(x^2)^5

Next calculate ((n),(r)) and simplify:

1+5x^2+10x^4+10x^6+5x^8+x^10

Short cut:

color(white)(0)^nC_(r)=color(white)(0)^nC_(n-r)

When one of the terms is 1, it is unnecessary to write out the powers. This was just done for completeness.