Cubic Roots

  1. 1. The problem statement, all variables and given/known data

    The equation:

    x^3 + ax + b = 0

    has 3 roots, u, p and q.

    Give the general solution for for an equation with roots (u/p)+(p/u), (p/q)+(q/p) and (u/q)+(q/u)

    2. Relevant equations

    u + p + q = 0

    upq = -b

    up + uq + pq = a

    If you can solve it you probably already knew those.


    3. The attempt at a solution

    Well I've just done lots of fiddling with algebra and got a pretty nasty looking solution, and I'm not even sure if it's right. What I was trying to do was express one of the new roots in terms of a, b and u, and then plug back in to the original equation for the new equation. I've got a pretty rough looking solution but I wanted to see if anyone could find a elegant way of solving this or if it is just lots of scruffy algebra. Thanks.
     
  2. jcsd
  3. rock.freak667

    rock.freak667 6,231
    Homework Helper

    You'll need to expand out

    [x-((u/p)+(p/u))][x-((p/q)+(q/p))][x-((u/q)+(q/u))]=0

    and then use the conditions given in your relevant equations. I don't think there is a simpler way.

    EDIT: I think you can say in general you will have Ax^3+Bx^2+Cx+D=0

    with the roots required, the sum will be -B/A and then you can just simplify the sum of the roots and get B/A and so on.
     
  4. Ah, that's what I was afraid of. Oh well, thanks. I guess it is just an ugly problem.
     
  5. rock.freak667

    rock.freak667 6,231
    Homework Helper

    Re-read my edit and see if that will help, I did not check to see if it will but it should work the same way.
     
  6. Yeah this was the first method I tried. Unfortunately I got stuck with the algebra and couldn't simplify it down any further, it was a very long horrid fraction. It's fine really though, doing the question wasn't really my interest, I just wanted to know if there was a shortcut through this problem really. Now it seems apparent that this question is just hard for the sake of being hard.
     
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