2009 UNCO Math Contest II Problems/Problem 7
A polynomial has a remainder of when divided by and a remainder of when divided by . What is the remainder when is divided by ?
Since we're being asked to find a remainder when a polynomial is divided by a quadratic, we can assume that the remainder will be at most linear. Thus, the remainder can be written in the form .
It is given that the polynomial has a remainder of when divided by and a remainder of when divided by , which translates to and . However, for both of these equations to always be true, the coefficient must be equal to .
Thus, and . These equations simplify to and , which shows that , so , meaning .
Plugging back into either equation gives , meaning the remainder is .
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