1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Viete relations problem

  1. May 21, 2008 #1
    [SOLVED] Viete relations problem

    1. The problem statement, all variables and given/known data
    Find all real numbers r for which there is at least one triple (x,y,z) of nonzero real numbers such that

    [tex] x^2 y + yz^2 + z^2 x = xy^2 + yz^2 + zx^2 = rxyz[/tex]

    2. Relevant equations

    3. The attempt at a solution
    This is equivalent to finding the possible values of r+s+t = 1/r + 1/s + 1/t where r,s,t are real but I don't see how that leads to a solution.

    Fix r and assume that x,y,z exist. Let f(t) = t^3 + at^2 + bt+c be the monic polynomial with
    x,y,z as its zeros. By assumption c is not zero. Its not hard to show that ab = (3+2r)c and a^3 = x^3+ y^3+z^3 + (3+2r)c using Viete's relations. But I am not sure what to do with those or how to get any sort of condition on r.

    Please just provide a hint.
  2. jcsd
  3. May 22, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Just to be sure, did you type out the equations correctly? Or is it supposed to be [itex]\sum x^2 y = rxyz[/itex] instead?
  4. May 22, 2008 #3
    I did mess up. Change yz^2 to y^2 z on the LHS. Anyway I already peeked at the solution.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook