Let $μ$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots, z_n$ be independent random variables, $z_i \sim μ$, and consider the random polynomial $$ p_n(z) = \prod_{k=1}^{n}{(z - z_k)}.$$ We determine the asymptotic distribution of $\left\{z \in \mathbb{C}: p_n(z) = p_n(0)\right\}$. In particular, if $μ$ is radial around the origin, then those solutions are also distributed according to $μ$ as $n \rightarrow \infty$. Generally, the distribution of the solutions will reproduce parts of $μ$ and condense another part on curves. We use these insights to study the behavior of the Blaschke unwinding series on random data.