Question

How do you write an equation for the terms in an answer for a squared polynomial with x terms?

For example `(a+b)^2 = a^2+2ab+b^2` the polynomial has two terms, the squared answer has 3 terms

along with the pattern `(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2cb` the polynomial has three terms and the squared answer has 6.
and the pattern keeps going. How do I write an equation to show if x terms in polynomial then y terms in answer?

so `(a+b+c+d.......)^2= x`number of terms

Answer 1

`1/2(n^2+n)`

Explanation:

`(overbrace(a_1+a_2+...+a_n)^"n terms")^2 = overbrace(a_1^2+a_2^2+...+a_n^2)^"n terms"+overbrace(2a_1a_2+2a_1a_3+...+2a_(n-1)a_n)^(""^nC_2 = 1/2(n^2-n) "terms")`

The total number of terms on the right hand side is:

`n+1/2(n^2-n) = 1/2(n^2+n)`

Essentially the number of terms on the right hand side is `n^2` counting multiplicity - that is, if you count each of the terms of the form `2a_ja_k` twice. Since we don't want to do that, we have to subtract `1/2(n^2-n)` from `n^2` to get `1/2(n^2+n)`.