It is given that A,B,C and D are roots of the quartic equation X^4 - X +2 = 0. Determine, whether or not (AB+CD) is a root of the equation X^3 – 8X –1 = 0.

(In reply to re: Nothing elegant yet by Richard)

The question does not say that A,B,C and D are distinct roots.

But if they are then we can let

A=r Exp( I alpha); B = r Exp (- I alpha);

C=s Exp( I beta); D = s Exp (- I beta);

giving the three possible perms X = AB+CD as

r^2+s^2, 2 r s Cos(alpha + beta) and 2 r s Cos (alpha - beta) which are all real.  The cubic has 3 real roots, one positive.

Some additional identities fall out of comparing the coefficients of the quartic and (X-A)(X-B)(X-C)(X-D) but so far nothing conclusive.

I suspect though from the way the question is written, that there is a different (more general) way of doing this problem.

 

 

Posted by goFish on 2005-12-23 05:23:58