2007 AIME I Problems/Problem 8

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Problem

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2+ (2k-43)x + k$ are both factors of $P(x)$?

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it. For them both to be factors of $P(x)$, they must both share a common factor. Denote this factor as $m$, and the other two factors $n$ and $o$.

Therefore, $(x - m)(x - n) = \ldots$ and $(x - m)(x - 0) = \ldots$.


The answer is $k = 30$.

See also

2007 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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