Solving a fifth degree polynomial

edit

Be specific on what kind of roots you want to find (rational ones? real ones? complex ones?)

sage: f=x^5-5*x^4-10*x^3-10*x^2-5*x-1
sage: f.roots(ring=RR)
[(6.72502395887258, 1)]
sage: f.roots(ring=CC)
[(6.72502395887258, 1),
 (-0.461764344593279 - 0.161600091968187*I, 1),
 (-0.461764344593279 + 0.161600091968187*I, 1),
 (-0.400747634843009 - 0.678737070411573*I, 1),
 (-0.400747634843009 + 0.678737070411573*I, 1)]
sage: f.roots(ring=QQbar)
[(6.725023958872576?, 1),
 (-0.4617643445932788? - 0.1616000919681873?*I, 1),
 (-0.4617643445932788? + 0.1616000919681873?*I, 1),
 (-0.4007476348430091? - 0.6787370704115728?*I, 1),
 (-0.4007476348430091? + 0.6787370704115728?*I, 1)]

We just see that the given polynomial is $2x^5-(x+1)^5$ and we are done.

If the computer is still needed, than the following lines investigate the factorization and the roots in sage:

sage: K.<z>  = CyclotomicField( 5 )
sage: RK.<X> = K[]
sage: F.<a>  = K.extension( X^5-2 )
sage: RF.<x> = F[]
sage: P = x^5 - 5*x^4 - 10*x^3 -10*x^2 - 5*x - 1

sage: for f in P.factor():    print f
(x + (z^3 + z^2 + z + 1)*a^4 - z^3*a^3 - z^2*a^2 - z*a - 1, 1)
(x - z^2*a^4 + (z^3 + z^2 + z + 1)*a^3 - z*a^2 - z^3*a - 1, 1)
(x - z*a^4 - z^2*a^3 - z^3*a^2 + (z^3 + z^2 + z + 1)*a - 1, 1)
(x - z^3*a^4 - z*a^3 + (z^3 + z^2 + z + 1)*a^2 - z^2*a - 1, 1)
(x - a^4 - a^3 - a^2 - a - 1, 1)

sage: P.roots( multiplicities=False)
[(-z^3 - z^2 - z - 1)*a^4 + z^3*a^3 + z^2*a^2 + z*a + 1,
 z^2*a^4 + (-z^3 - z^2 - z - 1)*a^3 + z*a^2 + z^3*a + 1,
 z*a^4 + z^2*a^3 + z^3*a^2 + (-z^3 - z^2 - z - 1)*a + 1,
 z^3*a^4 + z*a^3 + (-z^3 - z^2 - z - 1)*a^2 + z^2*a + 1,
 a^4 + a^3 + a^2 + a + 1]

In the above, $z$ is $\zeta=\zeta_5$, a primitive root of order $5$ of one. And $a$ is $2^{1/5}$.

The five roots are found manually by starting with the equation $2x^5=(x+1)^5$, dividing by $x^5$, so we get equivalently $\left(1+\frac 1x\right)^5=2$. So $\left(1+\frac 1x\right)^5=a\zeta^k$ for $k$ among $0,1,2,3,4$. This gives immediately: $$ x = -\frac 1{1-a\zeta^k} = \frac {1-(a\zeta^k)^5}{1-a\zeta^k} = 1 +a\zeta^k +a^2\zeta^{2k} +a^3\zeta^{3k} +a^4\zeta^{4k}\ . $$ One of the roots is thus $1+a+a^2+a^3+a^4$. For $k=0$. The others are obtained from it by replacing $a$ with a (conjugate) root of $X^5-2$.

Note:

If the coin is not immediately falling while typing the polynomial, then asking

sage: R.<x> = QQ[]
sage: P = x^5 - 5*x^4 - 10*x^3 -10*x^2 - 5*x - 1
sage: P.galois_group()

may help. I've got an error mess...
verbose 0 (2072: permgroup_named.py, cardinality) Warning: TransitiveGroups requires the GAP database package. Please install it with sage -i database_gap.

But instead of making it work, the following was enough for me:

dan ~$ gp
Reading GPRC: /etc/gprc ...Done.

PARI/GP is free software, covered by the GNU General Public License, and comes 
WITHOUT ANY WARRANTY WHATSOEVER.

?  P = 2*x^5 - (x+1)^5
%1 = x^5 - 5*x^4 - 10*x^3 - 10*x^2 - 5*x - 1
? polgalois( P )
%2 = [20, -1, 1, "F(5) = 5:4"]
? ?polgalois
polgalois(T): Galois group of the polynomial T (see manual for group coding). 
Return [n, s, k, name] where n is the group order, s the signature, k the index 
and name is the GAP4 name of the transitive group.

The order is making the group solvable already.