Prove $F(x, y) = \sum_{i=1}^{p-1} x^i y^{p-i} \equiv 0 \mod p$, for $x \ne y$ and prime $p$

Question

For $x=y$ the function just equals $-x$, and it is easy to see. I can't see why it is zero otherwise. Could you help me please?

Answer 1

Hint:

The claim is true if either $x,y\equiv_p 0$. Assume then that $x,y\not\equiv_p 0$.

$$F(x,y)=y^p\sum_{i=1}^{p-1}(xy^{-1})^i$$The sum is now a geometric series, can you take it from here? Can you see why the claim is false, if $x=y$?