Zero Sum of all Coefficients

The sum of the coefficients of $P(x)$ is nothing other than $P(1)$ (substitute $x=1$ everywhere — every power of $1$ is $1$, so what remains is exactly the sum of the coefficients). So:

If the coefficients of $P(x)$ sum to $0$, then $P(1)=0$, i.e. $1$ is always a root.

This hands over an immediate first factor $(x-1)$ to divide out, after which the reduced (lower-degree) equation is usually much easier. …