This math team problem is giving me a headache. It is one of 20 problems from the 1999 ICTM State Competition for Algebra II. The students have 50 minutes. I’ve spent hours on this one.

Given, , , Let be the remainder of , the y-intercept of which can be expressed as . Find .

I think there must be part of the remainder theorem I’ve either forgotten or never learned. I’ve backed into the solution^{1}, I just don’t know *why* I can do it.

Ideas?

^{1}The answer is in comments. I don’t want to ruin anyone’s fun.

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The answer is 1997, with a = 2, b = 3, and c = 1998.

Factor . Then we have

because the remainder must be a linear polynomial. Comparing constant terms, we must have

as the y-intercept you’re interested in. Any clearer yet?

That’s how I backed into the solution by testing , …

Thank you.

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