The best way to find the minimum of a polynomial (or, in general, any differentiable function) is to take the derivative and solve it for 0. This actually gives the extreme; you would then evaluate the 2d derivative at that point to see whether its a maximum or minimum. You could try Math::Symbolic::Derivative or Math::Derivative to get the derivatives. If you're only concerned with a range of x values, and f'(x) doesn't cross zero in this range, the minimum will be at one of the endpoints of the interval.

Since you're always dealing with polynomials, it should be simple to write a sub to give you the derivative, i.e., you can probably do without Math::Symbolic::Derivative or Math::Derivative.