6x6-25x5+31x4-31x2+25x-6=0,
6(x6-1)-25x(x4-1)+31x2(x2-1)=0,
6(x2-1)(x4+x2+1)-25x(x2-1)(x2+1)+31x2(x2-1)=0,
(x2-1)(6x4+6x2+6-25x3-25x+31x2)=0,
(x2-1)(6x4-25x3+37x2-25x+6)=0=(x-1)(x+1)(2x-1)(x-2)(3x2-5x+3).
Obtained by double synthetic division:
2 | 6 -25 37 -25 6
6 12 -26 22 | -6
½ | 6 -13 11 -3 | 0
6 3 -5 | 3
6 -10 6 | 0 = 2(3x2-5x+3)
The quadratic has no real factors, so the real x solution is:
x=-1, 1, ½, or 2. The reciprocal of x is also -1, 1, ½, 2.
Note that 6/x6-25/x5+31/x4-31/x2+25/x-6=0 has the same solution (reciprocal form).