The idea of the cubic polynomial generator is that for the A-Math syllabus, there must be 1 factor that is linear, and expressible in the form of Ax + B, for integer coeffiicients. As for the quadratic factor independent of the linear factor, it can either have 0, 1 or 2 real root (distinct or not). Hence, I first write a "quadratic generator", and then separately a "linear" generator". 

Putting them together gives me a cubic polynomial. This code generates a list of possible cubic polynomials with incredible precision, more often than not, and storing information as to what type of "roots" I expect to get. 

The inspiration was the fact that I found one of my "Worked Example" was unsolvable, which led me to the idea of Cubic Generator. This set of polynomials is what I use in my curriculum work as it is "error" free. 

A random genreator produces a linear factor that guarantees the existence of a rational root, because sometimes all roots can be irrational.

But we also need a quadratic root - whether factorizable, irrational or neither.

We now put them together!