Consider
$$\frac{(x-1)^y}{x} = z,\;y>2,\;z \neq 0$$
where $y$ is an integer.
In relation to solutions for $x$; How could one prove that:
$(1)$: There are $y$ solutions for $x$, in total.
$(2)$: If $y$ is odd, and $z$ is positive, there is $1$ real solution, and $y-1$ complex solutions.
$(3)$: If $y$ is even, and $z$ is positive, there are $2$ real solutions, and $y-2$ complex solutions.
$(4)$: If $y$ is odd, and $z$ is negative, there is $1$ real solution, and $y-1$ complex solutions.
$(5)$: If $y$ is even, and $z$ is negative, there are no real solutions, and $y$ complex solutions.