Solving Equation with Roots (u/p)+(p/u), (p/q)+(q/p), (u/q)+(q/u)

[jbar18]

Homework Statement

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)

Homework Equations

u + p + q = 0

upq = -b

up + uq + pq = a

If you can solve it you probably already knew those.

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 into 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.

 

[rock.freak667]

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.

 

[jbar18]

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.

Ah, that's what I was afraid of. Oh well, thanks. I guess it is just an ugly problem.

 

[rock.freak667]

Ah, that's what I was afraid of. Oh well, thanks. I guess it is just an ugly problem.

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.

 

[jbar18]

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.

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.