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Real roots of complex polynomials

  1. Feb 8, 2010 #1
    1. The problem statement, all variables and given/known data

    Let f be a polynomial of degree n >= 1 with all roots of multiplicity 1 and real on R. Prove that
    f has at most one more real root than f'
    f' has no more nonreal roots than f

    2. Relevant equations

    We are given the Gauss Lucas theorem: Every root of f' is contained in the convex hull of the roots of f.
    Also previously proved is that if zk is a root of multiplicity 1 of f(z) then f'(zk) !=0 (zk is not a root of the derivative of f)

    we express f(z)=c(z-z1)......(z-z2) where for each k f(zk)=0.

    3. The attempt at a solution

    I'm lost at where to begin, I've tried looking at it different ways but am not seeing where the difference of real and non real roots comes in.
    Any suggestions? (if you can help I'd prefer hints rather than the whole answer)

  2. jcsd
  3. Feb 8, 2010 #2
    R is a line in C, what can you say about the convex hull of collinear points?
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