# Ways to find irrational roots of an n degree polynomial

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.

• Numeric methods can be very interesting in their own right. Before using Newton methods and their close relations, separating roots (isolating them in the complex plane) is necessary in general. It would be more in keeping with your intent to "write a program" to learn more about numerical root finding algorithms. Jul 22, 2016 at 1:50
• You might want to look for the series of books by John McNamee. Jul 22, 2016 at 11:28

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
• @IanMiller: Roots are either rational or irrational. What Dr. Israel says is true, that the irrational roots generally cannot be expressed using only symbols for radicals and rational arithmetic, once degree $n \ge 5$. Jul 22, 2016 at 2:19
• To expand on Robert's note: it is known that there are algebraic numbers of degree 5 and higher that cannot be expressed in terms of radicals; one thus has to use special functions like the hypergeometric functions, Meijer $G$, (multivariate) theta functions, or (hyper)elliptic functions to represent them in closed form. Jul 22, 2016 at 11:26