1999 AHSME Problems/Problem 17
Contents
Problem
Let be a polynomial such that when is divided by , the remainder is , and when is divided by , the remainder is . What is the remainder when is divided by ?
Solution 1
According to the problem statement, there are polynomials and such that .
From the last equality we get .
The value is a root of the polynomial on the right hand side, therefore it must be a root of the one on the left hand side as well. Substituting, we get , from which . This means that is a root of the polynomial . In other words, there is a polynomial such that .
Substituting this into the original formula for we get
Therefore when is divided by , the remainder is .
Solution 2
Since the divisor is a quadratic, the degree of the remainder is at most linear. We can write in the form where is the remainder. By the Remainder Theorem, plugging in and gives us a system of equations.
Solving gives us and , thus, our answer is
See also
1999 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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