Hey petenice.

Factoring is a weird thing: in root finding a wide range of approaches are taken.

One that is commonly taught is to "guess" one root and then factor the polynomial by dividing P(X) by (x-a) where a is the root. It's not a mechanical systematic thing unless you have previous experience or can "see" something in the polynomial itself (i.e. an easy solution).

There is however a formula for obtaining the roots to a cubic polynomial given a cubic equation and it is more complex than the standard quadratic.

You also have other results in algebra that give ways to factorize equations of a particular form.

The cubic formula gives you a systematic way to get the roots and the proof shows you how the routine is constructed, but when you don't have these and try to "guess" roots then that's what happens: you make a few guesses and if you get one right then you do the long division with a polynomial and you factorize out to solve something simpler.