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Homework Help: F r is a rational solution r = p/q and p and q are coprime, show that q|an and p|a0

  1. Sep 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose that r is a solution of the equation:

    anxn + a(n−1)x(n−1) + . . . + a1x + a0 = 0

    where the coefficients ak belongs to Z for k = 0, 1, . . . n, and n is greater or equal to 1. If r is a rational solution r = p/q, where p, q belong to Z and p and q are
    coprime, show that q|an and p|a0.


    3. The attempt at a solution

    Im not even sure where to begin, im so confused, what am i trying to prove? and how do i prove it, i feel like there is something missing in the question.
     
  2. jcsd
  3. Sep 29, 2010 #2

    Mark44

    Staff: Mentor

    Re: f r is a rational solution r = p/q and p and q are coprime, show that q|an and p|

    Here's an example to help show you how this works. Here's an equation: x2 - 4x + 4 = 0.

    In this equation a2, the coefficient of x2, is 1. a0 is the constant term, and is 4.

    If there is a rational number r = p/q that is a solution to this equation, this theorem says that p has to divide a0, and q has to divide a2.

    As it turns out, 2 is a solution, and is a rational number - i.e., 2 = 2/1. Clearly 2 divides 4, and 1 divides 1.
     
  4. Sep 29, 2010 #3
    Re: f r is a rational solution r = p/q and p and q are coprime, show that q|an and p|

    Thanks i got it now, just got confused with the wording.
     
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