Last day of 2015 problem

Algebra Level 5

Sriram says to Hari, " I am thinking of polynomial whose roots are all positive integers. The polynomial has the form \(P(x)=2x^3-2ax^2 +(a^2-81)x-c\) for some positive integers \(a\) and \(c\). Can you tell me the values of \(a\) and \(c\) ? "

After some calculations, Hari says, " There is more than one such polynomial."

Sriram says, " You're right. Here is the value of \(a\)." He writes a positive integer and asks, " Can you find the value of \(c\) ? "

Hari says, " There are still two possible values of c ."

Find the sum of two possible values of c.


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