O15-C922
> restart:with(linalg):
Warning, new definition for norm
Warning, new definition for trace
> 
> base:=[seq(t^k,k=0..4)];

                                      2   3   4
                      base := [1, t, t , t , t ]

> image:=[seq(expand(x^4*subs(t=x+1/x,base[k])),k=1..5)];

             4   5    3   6      4    2   7      5      3
  image := [x , x  + x , x  + 2 x  + x , x  + 3 x  + 3 x  + x,

         8      6      4      2
        x  + 4 x  + 6 x  + 4 x  + 1]

> m:=matrix(5,5,[0,0,0,0,1,0,0,0,1,0,0,0,1,0,4,0,1,0,3,0,1,0,2,0,6]);

                          [0    0    0    0    1]
                          [                     ]
                          [0    0    0    1    0]
                          [                     ]
                     m := [0    0    1    0    4]
                          [                     ]
                          [0    1    0    3    0]
                          [                     ]
                          [1    0    2    0    6]

> n:=evalm(m^(-1));

                        [ 2     0    -2    0    1]
                        [                        ]
                        [ 0    -3     0    1    0]
                        [                        ]
                   n := [-4     0     1    0    0]
                        [                        ]
                        [ 0     1     0    0    0]
                        [                        ]
                        [ 1     0     0    0    0]

> q:=transpose(matrix(1,5,[1,0,-11,-10,0]));#Le coeff de X^4 est 0 et non -5

                                   [  1]
                                   [   ]
                                   [  0]
                                   [   ]
                              q := [-11]
                                   [   ]
                                   [-10]
                                   [   ]
                                   [  0]

> r:=evalm(n&*q);

                                   [ 24]
                                   [   ]
                                   [-10]
                                   [   ]
                              r := [-15]
                                   [   ]
                                   [  0]
                                   [   ]
                                   [  1]

> p:=sum(r[k+1,1]*X^k,k=0..4);

                                          2    4
                     p := 24 - 10 X - 15 X  + X

> rac:=solve(p);

                         rac := 1, -3, -2, 4

> seq([solve(x+1/x=rac[k])],k=1..4);

  [1/2 - 1/2 I sqrt(3), 1/2 + 1/2 I sqrt(3)],

        [-3/2 + 1/2 sqrt(5), -3/2 - 1/2 sqrt(5)], [-1, -1],

        [2 + sqrt(3), 2 - sqrt(3)]

>