Let
$f_n(x)=\prod\limits_{i=1}^n (x+i)-n!=(x+1)(x+2)\cdots(x+n)-n!$
$n$ is a positive integer. What are the roots of the polynomial for a given $n$ except $0$? Or determine the real part of the roots or even determine the maximum real part of the roots except $0$.
Thanks!
There are always $n$ complex solutions to degree $n$ polynomial, thanks to the fundamental theorem of algebra. I will only provide real solutions here. Note that:
For $x>0$, the value is always greater than zero.
For $x<-n-1$, the absolute value is always greater than zero.
For $0>x>-n-1$, the value can't be zero.
Since $x=0$ is one real root and $x=-n-1$ is the other real root if $n$ is even, the polynomial has one or two real roots.
Attached are cases when $n=3$ and when $n=4$:
