1. Background/theory We know that if the equation x3+px2+qx+r=0 has solutions x1, x2, x3 then x1 + x2 + x3 = -p x1x2 + x2x3 + x3x1 = q x1x2x3 = -r 2. Problem statement Find (x1 - x2)2(x2 - x3)2(x3 - x1)2 as an expression containing p,q,r. That is, I'm supposed to find the discriminant of the above cubic equation in terms of its coefficients. 3. The attempt at a solution Now, I have approached the problem in two ways. First, I expanded the whole expression out and tried to manipulate the expression to get it in the desired terms. It is a very lenthy and laborious process though and I had trouble getting the right expression. So there must be some more elegant solution to this problem than just multiplying the expression out and manipulate it. My best attempt at a "more elegant" solution was to reason as follows: When we expand the expression (x1 - x2)2(x2 - x3)2(x3 - x1)2 we get a polynomial with terms of degree 6 in x1, x2, x3. Now, p has degree 1, q has degree 2 and r has degree 6 so the possible terms in the expression is: p6, q3, r2, pqr, p3r, p4q, p2q2. So we can write: (x1 - x2)2(x2 - x3)2(x3 - x1)2 = Ap6 + Bq3 + Cr2 + Dpqr + Ep3r + Fp4q + Gp2q2 where A,...,G are constants. Then I tried to determine A,...,G by finding cubic polynomials with given roots and thus given values of the discriminant and p, q, r. I easily determined the terms -27r2 and -4q3 by just letting p=0 but when I tried to determine the other constants I got wierd numbers like A=770,4 and G=87,7 etc, whereas I expected those constants to be zero (since the discriminant does not contain p2q2, for instance). I can't explain why... I am grateful for any guidance here! Also, I apologize for possible language errors, English is not my native.