For the polynomial $P = x^2 + 3x + 1$ $\in Z [x]$, determine the zeros of $P$ modulo $11$, i.e. in $Z/11Z$, as well as at least one zero modulo $11^2 = 121$, i.e. in $Z/121Z$.
I see, there is a polynomial degree 2, so i know the polynomial has a maximum of two zeros.
Trying P modulo 11, for P(0-6) we get zeros in $P(2) = 2^2+3*2+1 = 11 mod 11 = 0$ and $P(6) = 55 mod 11 = 0$ , with two zeros, i can stop now.
But how i have to solve the task for $Z/121Z$ is there a faster way without trying it for every number from P(0-120)?