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Aime 2007 I 8

  1. May 26, 2008 #1
    1. The problem statement, all variables and given/known data
    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)?

    2. Relevant equations

    3. 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!?
  2. jcsd
  3. May 26, 2008 #2


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    Staff Emeritus
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    Presumably you only need to know that it is cubic, so take some general cubic function.
  4. May 26, 2008 #3
    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!
  5. May 26, 2008 #4


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    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.
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