For polynomial lovers!

Algebra Level 5

There are nonzero integers \(a,b,r\) and \(s\) such that the complex number \(r+si\) is a zero of the polynomial \(P(x)=x^3-ax^2+bx-65\). For each possible combination of \(a\) and \(b\), let \(p_{a,b}\) be the sum of zeros of \(P(x)\). Find the sum of the \(p_{a,b}\)'s for all the possible combinations of \(a\) and \(b\).

Here \(i=\sqrt{-1}\)

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