Proving a formula

  1. 1. The problem statement, all variables and given/known data
    Prove that if p(x)=anx^n +an-1x^n-1+..........a0, where a0,.........., "an" ε reals, is a polynomial, then p can have at most n roots.

    2. Relevant equations

    3. The attempt at a solution

    C ε R is a root of a polynomial p if p(c)=0. If c is a root of p, then x-c is a factor of p.

    I'm not sure where to go from here. I think it would probably be the easiest to prove this by proving the contrapositive as being false. Could someone please give me a hint or show me where to go from here?

    Thank you very much
  2. jcsd
  3. Dick

    Dick 25,735
    Science Advisor
    Homework Helper

    Sure. Suppose the polynomial has n+1 different roots. c1,c2, Since c1 is a root the polynomial p(x) can be factored (x-c1)*p1(x) where p1 has degree n-1. The other c's must be roots of p1(x) since they aren't roots of (x-c1). Continue in this way until you reach degree 1. Now you have a linear polynomial with two different roots. Possible?
  4. Thank you very much

    Would it be somthing like this?

    (p1x)^(n-1)(x-c2)(x-c3)^(n) :confused:

    Thank you
    Last edited: Mar 2, 2008
  5. Dick

    Dick 25,735
    Science Advisor
    Homework Helper

    No, that's not clear at all. Start by proving if n=1 then the polynomial can't have 2 roots. Ok?
  6. Thank you very much

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