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.

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