How to solve $2z^5 + z^4 -6z^2 + z + 1 = 0$?

Question

$$2z^5 + z^4 -6z^2 + z + 1 = 0$$
z is complex number.
I tried to make factor but i didn't find.

Answer 1

Using a CAS, the Galois group of this polynomial is $S_5$, so the roots can't be expressed using radicals. You need to use numerical methods.

Input:

Q := Rationals();
P<z> := PolynomialRing(Q);
G, R, S := GaloisGroup(2*z^5+z^4-6*z^2+z+1);
G;

Output:

Symmetric group G acting on a set of cardinality 5
Order = 120 = 2^3 * 3 * 5