1) x= 0 and 1 are not roots
2)
f(x)= x^n + 2 x^{n-1} + ...+ nx + n+1
\frac{f(x)}{x} = x^{n-1} + 2x^{n-2} +....+ n + \frac{n+1}{x}
Thus f(x)- \frac{f(x)}{x} = \frac{x^{n+1}-1}{x-1} - \frac{n+1}{x} ________(1)
3) If x_{k},\; k\; = \left\{1,2,3..n \right\} are roots of f(x) = 0 we have from (1)
\sum_{k=1}^{n}{}\frac{x_{k}^{n+1}-1}{x_{k}-1} = \sum_{k=1}^{n}{}\frac{n+1}{x_{k}} = -n
Note that \sum_{k=1}^{n}{}\frac{1}{x_{k}}= \; -\frac{n}{n+1}