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> Later on, I reduced another interviewer's search problem to a polynomial equation that had integer roots when a solution existed

well, you can reduce any computable problem to that form

see http://en.wikipedia.org/wiki/10th_Hilbert_problem

For the single variable case there is an algorithm: http://en.wikipedia.org/wiki/Rational_root_theorem

Whether it is a fast algorithm depends heavily on the sizes of the first and last coefficients, as you have to factorize them.

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