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Polynomial with at most n-1 solutions.

  1. Jan 22, 2013 #1
    Hi,
    1. The problem statement, all variables and given/known data
    I am expected to show that the polynomial
    a1xb1 + a2xb2 + ... + anxbn = 0
    has at most n-1 solutions in (0,infinity), where a1,a2,...,an are real numbers different than zero, and b1,b2,...,bn are real numbers so that bj is different than bk for j different than k.


    2. Relevant equations



    3. The attempt at a solution
    I am trying to apply Rolle's theorem, but am not very successful at that. In general, between any two solutions the first derivative is equal to zero.
    I first tried dividing by xb1 noting that zero is not a solution and in order to obtain a simpler polynomial, but it is doubtful this is how it ought to be solved and it didn't seem to get me anywhere.
    Would anyone kindly provide some further insight/guidance?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 22, 2013 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Try thinking about the cases n=1 and n=2 first. You've got some good ideas there of using Rolle's theorem and dividing by x^(b1). Try and apply them to set up a proof by induction.
     
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