# [Maxima] Finding roots of sextic in radicals

Alexey Beshenov al at beshenov.ru
Tue Jan 22 04:31:51 CST 2008

```On Tuesday 22 January 2008 07:04, Jordi Gutiérrez Hermoso wrote:

> If my calculations are correct, the roots of x^6 + 3*x^5 + 6*x^4 +
> 3*x^3 + 9*x + 9 should all be expressible by radicals. In fact, they
> are all polynomials in terms of 2^(1/3) and a cube root of unity.
>
> How can I make Maxima tell me what the roots actually are?

There's no general solution in radicals to polynomial equations of degree 5 or
higher, so it seems that you should use numerical methods:

(%i1) allroots (x^6 + 3*x^5 + 6*x^4 + 3*x^3 + 9*x +9);
(%o1) [x = 0.86602540378444 %i + 0.75992104989487,
x = 0.75992104989487 - 0.86602540378444 %i,
x = 0.22509823218729 %i - 1.129960524947437,
x = - 0.22509823218729 %i - 1.129960524947437,
x = 1.95714903975616 %i - 1.129960524947437,
x = - 1.95714903975616 %i - 1.129960524947437]

--
Alexey Beshenov <al at beshenov.ru>
http://beshenov.ru/
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