Show that the given polynomial has distinct roots.

Question

Show that for each $k$ the equation $z (z-1)(z-2) \cdots (z-n+1) = k$ has all it's roots distinct.

How should I proceed? Please help me. Can I do it by taking derivative as I have observed that it's derivative has all it's roots distinct. How does it help in solving this problem?

Please give me some hint.Thank you very much.

Answer 1

$z(z-1)(z-2)(z-3)+1 = (z^2-3z+1)^2$ has repeated roots.