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.

  • 2
    $\begingroup$ Is $k$ an integer? You didn't specify anything about $k$. And if $k$ can be any real number, then the statement isn't true: $z(z-1)=-\frac{1}{4}$ has a repeated root. $\endgroup$ – zipirovich Mar 20 '18 at 18:42
  • $\begingroup$ OK! Assume $k$ to be an integer.Then how should I argue? $\endgroup$ – Dbchatto67 Mar 20 '18 at 18:43

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

  • $\begingroup$ Actually this question has been given to my assignment.Is it then wrong? $\endgroup$ – Dbchatto67 Mar 20 '18 at 18:50
  • 2
    $\begingroup$ @DebabrataChattopadhyay: it is clearly wrong. By considering $f(z)=z(z-1)\cdots(z-(2m+1))$ we have that $z=m+\frac{1}{2}$ is a double root for $f(z)=f(m+1/2)$, by symmetry. $\endgroup$ – Jack D'Aurizio Mar 20 '18 at 20:15
  • $\begingroup$ Thanks @Jack D'Aurizio. $\endgroup$ – Dbchatto67 Mar 21 '18 at 1:29

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