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Sum of All Possible Sums of Roots of Cubic

Source: 2013 AIME I Problem 10

March 15, 2013

algebrapolynomialAMCAIMEAIME I

Problem Statement

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 the zeroes of

P(x)

. Find the sum of the

p_{a,b}

's for all possible combinations of

a

and

b

.

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