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

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Answer 1 · 2018-03-07T19:41:03

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

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.