I am trying to write a program to find the roots a given polynomial of degree N, with the form
$$
A_{0}X^{N}+A_{1}X^{N-1}+A_{2}X^{N-2}+A_{3}X^{N-3}+…+A_{N}
$$
I know that if there are rational roots at all, I can find an exhaustive list with the rational root theorem, and then factor them out using synthetic division to find any and all rational roots. I also know that I am fine if I can factor down to degree two, but I would like to know how to find the irrational roots of an nth degree polynomial without numeric ways like Newton's method, to be able to display the polynomial thusly.
$$
(x+2)(x-6)(x\pm\sqrt{8})…
$$
Any help to be had would be appreciated.
Best Answer
It can't be done. There are formulas for the roots of a quadratic, cubic or quartic in terms of radicals, but not (in general) for the roots of a polynomial of degree $5$ or higher. For example, the roots of $x^5 + 2 x + 1$ can't be written in terms of radicals. See e.g. Abel-Ruffini theorem