Ways to find irrational roots of an n degree polynomial

Question

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.

Answer 1

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

Answer 2

An efficient algorithm to factor a polynomial into irreducible polynomials is given in this article. The lattice basis reduction algorithm they developed for this purpose is the famous LLL algorithm which has many applications besides its use in polynomial factorization problems.