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Homework Help: Partial Fractions Question

  1. Sep 6, 2007 #1
    f(x) is a polynomial. A product of n distinct factors


    Prove that


    This I can do by writing f(x)=(x-a)g(x) where g(a)<>0. Then splitting




    for some h(x) so


    and differentiating f(x), f'(a)=g(a) so


    and I repeat that for each factor so I get the sum required.

    The next question I can't answer:

    Show that


    is 0 when r=0,1,...,n-2 and is 1 when r=n-1

    I've tried expanding f(x), differentiating and putting x equal to the roots and summing over all the roots so I see the result works but I'm having no success in proving it.
    Could someone help with a hint or two?
    Last edited by a moderator: Sep 6, 2007
  2. jcsd
  3. Sep 6, 2007 #2


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    What do you get when n = 2, say?
  4. Sep 6, 2007 #3


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    You might consider using logarithmic differentiation.

    If [itex]f(x)= a(x-x_1)(x- x_2)\cdot\cdot\cdot(x- x_n)[/itex]
    then [itex]ln(f(x)) ln a+ ln(x-x_1)+ \cdot\cdot\cdot+ ln(x-x_n)[/itex]
    what do you get when you differentiate that?
  5. Sep 7, 2007 #4
    Yes I see that the result works for n=2 and even 3. But I do not see how that helps starting off a proof by Induction.
  6. Sep 7, 2007 #5
    Yes I tried this too but I still cannot see how I get the [tex](a_{i})^r[/tex] terms unless I do something like what I tried in my first post i.e. use an expanded form of f(x) and differentiate as the highest order term is of order n-1.
  7. Sep 7, 2007 #6


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    How does 1/f(x) look if you use partial fractions... get a few terms... you'll see a pattern...

    Then examine what f'(ai) looks like.

    Relate the above 2 ideas...
  8. Sep 8, 2007 #7
    Thanks very much for the help.
    I see it now: Consider a1^r/f'(a1) as the polynomial x^r/[(x-a2)(x-a3)...(x-an)] and split this into partial fractions just like 1/f(x) above and you find a1^r/f'(a1) is just he negative of the sum of all other such terms with the other n-1 factors of f(x).
    This works for r<n-1. For the r=n-1 case I differentiated the expanded f(x), divided by f'(x), and then substituted in a1,a2,...,an and added. From above all terms with r<n-1 are zero so we are left with n=n*Sum(a^(n-1)/f'(a)).

    Thanks again for the hints.
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