2013 AIME I Problems/Problem 10
There are nonzero integers , , , and such that the complex number is a zero of the polynomial . For each possible combination of and , let be the sum of the zeros of . Find the sum of the 's for all possible combinations of and .
Since is a root, by the Complex Conjugate Root Theorem, must be the other imaginary root. Using to represent the real root, we have
Applying difference of squares, and regrouping, we have
So matching coefficients, we obtain
By Vieta's each so we just need to find the values of in each pair. We proceed by determining possible values for , , and and using these to determine and .
If , so (r, s) =
Similarly, for , so the pairs are
For , so the pairs are
Now we can disregard the plus minus signs for s because those cases are included as complex conjugates of the counted cases. The positive and negative values of r will cancel, so the sum of the for is times the number of distinct values (as each value of generates a pair ). Our answer is then .
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