# [Maxima] Finding roots of sextic in radicals

S. Newhouse sen1 at math.msu.edu
Tue Jan 22 06:38:52 CST 2008

```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?
>
> - Jordi G. H.
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> Maxima at math.utexas.edu
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>
>

Using the fact that the roots are polynomials in 2^(1/3),
the following commands makes the list
sol_list
of roots.

The list
sol_test
tests that they are indeed the roots.

p:x^6+3*x^5+6*x^4+3*x^3+9*x+9;
p(i):=part(factor(p,q^3-2),i);
p(i):=part(factor(p,q^3-2),i);
s(i):= solve(subst(q=2^(1/3),p(i)),x);
r(i,j):= part(s(i),j,2);
sol_list: [r(1,1),r(1,2),r(2,1),r(2,2), r(3,1),r(3,2)];
test(i,j):= ratsimp(ev(subst(x = r(i,j),p)));
sol_test: [test(1,1),test(1,2),test(2,1),test(2,2),test(3,1),test(3,2)];

-sen
```