Do I understand the abel-ruffini theorem correctly?

Question

I am writing a program which relies heavily on solving complex equations. Of course I can't just write a separate function for solving every single type of equation, because I might as well need to solve an equation of 200th degree and I don't have the time to create 200 separate functions for every degree.

I was looking for a general formula for solving any equation with a single variable. The catch is that I'd much rather have the exact solutions, not approximated (or at least as exact as a finite set of bits allows them to be). Then I stumbled upon the Abel-Ruffini theorem.

So basically the way I understand it is that no such formula exists, that for an equation of degree >= 5 you either need to approximate the solutions or create a formula that is specific to that equation. Which basically means that what I was trying to achieve is not possible. But that just doesn't seem right. How come a human can easily solve something like ax^{10} = 0 or ax^8 + bx^4 = 0 but a machine can only approximate since it relies on having a formula to use? And why is number 5 so special? Do I even understand it correctly?

Answer 1

You misunderstand a vital part of the Abel-Ruffini theorem. It more or less states that polynomials of degree $>4$ are not solvable in general using radicals, but on a case-by-case scenario, they can be solved using radicals. Your examples are plenty proof of the fact.

For polynomials degree $5,6$, general formulas do exist, they just involve crazy functions like the hypergeomtric series. See here for the $5$th degree and here for the $6$th degree. Particularly, you will need there Kampé de Fériet function.

I also recommend you check the rational roots theorem, it helps reduce the problem often, and if you learn Galois theory, you can determine which polynomials of degree $>4$ are solvable with radicals.

Answer 2

A computer can certainly solve $ax^{10}=0$ and $ax^8+bx^4=0$ because those are special polynomials with a certain form. What it can't solve is the following general equation in terms of radicals: $$ax^5+bx^4+cx^3+dx^2+ex+f=0$$ There are certain special cases of this equation where some of the coefficients are $0$ or related to each other where it can be solved exactly, but the general case can not be solved using one formula. Also, what I mean by "in terms of radicals" is that you can not solve this equation by using addition, subtraction, multiplication, division, and roots in terms of $a, b, c, d, e, f$ to solve this equation. It's not just that there's no general formula with radicals. It's that it is literally impossible to write the roots of certain quintics using only radicals. The exact solution simply can not be written in terms of radicals. Even if you tried to create a specific formula just for $x^5+2x^3-19$ in terms of radicals, you can't. You have to use something else like maybe some trigonometric or other weird function in order to write the exact solution.

You can, however, reduce the quintic to Bring-Jerrard form (although, to be honest, I don't know how to do this reduction) and then use the Bring radical to solve the equation. If I'm not mistaken, the Bring radical can be solved using binary search since $x^5+x+a$ is an always increasing function, so I think you should still be able to approximate the solution to a quintic pretty well by approximating the Bring radical.