Determine if it is possible that p(x)*q(x)= x⁵+9x+1, if p(x) and q(x) must be polynomials of degree 1 or higher, with integer coefficients.

The degrees of p and q must be either 1 and 4 or 2 and 3.
The given polynomial has one real root which appears irrational which rules out a degree of 1.

So let p(x) = ax^2 + bx + c
and q(x) = dx^3 + ex^2 + fx + g

It is clear that d = a = +/- 1 and c = g = +/- 1

From here one can create a system of equations relating the rest of the variables.  This system, which I dont have time to explain right now, has no solutions.

Posted by Jer on 2007-03-13 09:34:49