I'm doing a homework problem where I have to find all roots of $(z+1)^7 - (z)^7 = 0$ using the roots of unity for $z^7$
I noticed that if $a$ is a root of unity for $z^7$, then $1/(a-1)$ maps the roots of $z^7$ to the roots of $(z+1)^7 - (z)^7$. I also noticed that this works for any power $n$, not just $n=7$. Similarly if we are solving $(z+c)^n - (z)^n = 0$, the solution appears to be $c/(a-1)$ where $a$ is the roots of unity for $n$.
furthermore, it seems for a polynomial $p(x)$, there is a conformal mapping between the roots of $p(x+1) - p(x)$ and the roots of $p(x)$
I discovered all of this while on mathematica and would appreciate an explanation to what is going on?
FYI this is my first post here!!