TITLE: Does this polynomial have all its roots both distinct and real?
QUESTION [13 upvotes]: Recently, I wondered about the following problem: let $n\geq 5$ and let
$$
P_n(x)=(x-1)(x-2)\ldots (x-n)-1
$$
Is it true that $P_n(x)$ has $n$ distinct real roots for any
$n\geq 5$ ? I checked it for up to $n=50$. I found nothing else so far.

REPLY [14 votes]: What happens when you evaluate this at $x=0, 1, 1.5, 2, 2.5 \dots$? Then use the intermediate value theorem. 
At the integer points the polynomial evaluates to $-1$. At alternate intermediate points the  value is positive, and you get a pair of roots.
If $n$ is even you have positive values at $x=0$ and $x=n+1$ so roots at "either end" to make the full complement.
If $n$ is odd you have a positive value at $x=n+1$ which does the same.