# Aime 2007 I 8

## Homework Statement

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)?

## The Attempt at a Solution

I don't understand the question. How can you determine whether Q_1 and Q_2 are factors of P(x) when they do not tell you what P(x) is!?

## Answers and Replies

cristo
Staff Emeritus
Science Advisor
Presumably you only need to know that it is cubic, so take some general cubic function.

How could the answer possibly not depend on what P(x) is? If not, and the answer is greater than 1, that implies that ALL cubic polynomials have a common factor which is absurd!

morphism
Science Advisor
Homework Helper
I agree - the wording could have been better. I think what they want you to do is assume that Q_1 and Q_2 are divisors of P, find the values of k for which this is possible, and give them the largest of these values. Using this interpretation, all you need to know about P is that it's a cubic.