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.

The quartic equation calculator http://www.1728.com/quartic.htm
nicely solves X^4-x+2=0.

By expanding out (X-A)*(X-B)*(X-C)*(X-D) and using that the roots are in complex conjugate pairs, it is not hard to show that
A*B+C*D=(2*Re(A))^2, and using the found root values, this quantity does satisfy X^3-8*X-1=0, to within numerical accuracy.

The proposer probably really wants an exact, purely algebraic, solution, however.

Posted by Richard on 2005-12-22 14:48:37