For $-1 < \alpha \leq 0$ and $0 < p < \infty$, the solutions of certain extremal problems are known to act as contractive zero-divisors in the weighted Bergman space $\Apa$. We show that for $0 < \alpha \leq 1$ and $0 < p < \infty$, the analogous extremal functions do not have any extra zeros in the unit disk and, hence, have the potential to act as zero-divisors. As a corollary, we find that certain families of hypergeometric functions either have no zeros in the unit disk or have no zeros in a half-plane.