  Let $\mu$ and $\nu$ be probability measures in the complex plane, and let $p$
and $q$ be independent random polynomials of degree $n$, whose roots are chosen
independently from $\mu$ and $\nu$, respectively. Under assumptions on the
measures $\mu$ and $\nu$, the limiting distribution for the zeros of the sum
$p+q$ was by computed by Reddy and the third author [J. Math. Anal. Appl. 495
(2021) 124719] as $n \to \infty$. In this paper, we generalize and extend this
result to the case where $p$ and $q$ have different degrees. In this case, the
logarithmic potential of the limiting distribution is given by the pointwise
maximum of the logarithmic potentials of $\mu$ and $\nu$, scaled by the
limiting ratio of the degrees of $p$ and $q$. Additionally, our approach
provides a complete description of the limiting distribution for the zeros of
$p + q$ for any pair of measures $\mu$ and $\nu$, with different limiting
behavior shown in the case when at least one of the measures fails to have a
logarithmic moment.
