Consider I have two degree $d$ polynomials $f_1$ and $f_2$, and they do not have any root in common. I need to compute $f_3=f_1+f_2$, but $f_3$ may have some roots in $R$. So I pick two random polynomials, $r_1,r_2$ of degree $d$, where $r_1$ and $r_2$ have no roots in common in $R$. Then I compute $f'_3=r_1\cdot f_1+ r_2+f_2$. My goal is to eliminate the possibility of addition of two polynomials to have an root in $R$. edit: all polynomials are defined over polynomial ring $R[x]$ where $R$ is $\mathbb{Z}_p$ and $p$ is a large prime number.
Question: What is the probability of $f'_3$ to have a root in $R$?