2007 AIME I Problems/Problem 8

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 $k = 30$ .

2007 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2007 AIME I Problems/Problem 8

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Solution

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