The equation x

^{5} –23x

^{3} –66x

^{2} +px +q =0 has three non-zero roots, and two of them are integers, with multiplicity 2.

Find p and q and determine all the roots of the given equation.

(In reply to

Who needs p and q? (SPOILER) by Old Original Oskar!)

It seems that there is a minor typographical error in terms of the solution to the above problem.

In terms of the remainder theorem, the polynomial P(x) = x^5-23x^3-66x^2-68x+24, possesses the respective values -256 and -132 corresponding to x=2 and x=1, not giving any zeroes for the aforementioned x values.<o:p></o:p>

Incidentally, ((x-2)^2)*((x-1)^2)*(x+6)= x^5 - 23x^3 +66*x^2- 68x + 24, so the coefficient of x^2 in P(x) should be +66 instead of -66.

*Edited on ***May 31, 2006, 2:42 pm**