# [Maxima] Finding roots of sextic in radicals

Andrej Vodopivec andrej.vodopivec at gmail.com
Thu Jan 24 15:07:57 CST 2008

```On Jan 22, 2008 5:04 AM, Jordi Gutiérrez Hermoso <jordigh at gmail.com> 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?

I don't know how to compute the roots in radicals, but I have some
code which can sometimes guess how they look like from their numerical
values:

(%i2) x^6 + 3*x^5 + 6*x^4 + 3*x^3 + 9*x + 9\$
(%i3) allroots(%)\$
(%i4) identify(%);
(%o4) [x=(sqrt(3)*%i)/2+(3*2^(1/3)-3/2)/3,x=(3*2^(1/3)-3/2)/3-(sqrt(3)*%i)/2,x=.2250982321872863*%i+(-3/4^(1/3)-3/2)/3,x=(-3/4^(1/3)-3/2)/3-
.2250982321872863*%i,x=1.95714903975616*%i+(-3/4^(1/3)-3/2)/3,x=(-3/4^(1/3)-3/2)/3-1.95714903975616*%i]
(%i5) subst(first(%), %o2), ratsimp; <-- you need to check if we
guessed correctly
(%o5) 0

After you have two roots written with radicals you should be able to
solve the polynomial of degree 4 to get others.

identify works mostly like the identify from maple but I think is less
powerfull. If you would find it interesting let me know.

--
Andrej

```