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Multiplicities of zeroes

  1. Nov 14, 2007 #1
    1. The problem statement, all variables and given/known data
    How do you show that if f(t) has a zero t_0 of multiplicity n, then f'(t) has a zero at t_0 of multiplicity EXACTLY n -1?

    I can prove that the multiplicity is at least n-1 but I am having trouble with the exactly part...

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Nov 14, 2007 #2
    if f(t) has a root of multiplicity n, then we can write f(t) = (t - t_0)^n g(t) where g(t_0) != 0. Then f'(t) = (t-t_0)^(n-1)k(t) where k(t) = ng(t) + (t-t_0)g'(t) and k(t_0) = ng(t_0) != 0, so f' has a root of multiplicity n-1.
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