z^(4) \- 5(z-1)\[z^(2) \- z + 1\] = z^(4) \- 5(z-1)\[z^(2) \- (z - 1)\] = z^(4) \- 5z^(2)(z - 1) + 5(z - 1)^(2) = \[z^(2) \- 5(z-1)/2\]^(2) \- 5(z-1)^(2)/4,

and you can factor the last expression as a difference of squares.

Edit: Alternatively, if you'll accept roots in trigonometric form, you can rewrite the expression as \[(z-1)^(5) \+ 1\]/z.