  In a previous work, we introduced the Collatz polynomials; these are the
polynomials $\left[P_N(z)\right]_{N\in\mathbb{N}}$ such that
$\left[z^0\right]P_N = N$ and $\left[z^{k+1}\right]P_N =
c\left(\left[z^k\right]P_N\right)$, where $c:\mathbb{N}\rightarrow \mathbb{N}$
is the Collatz function $1\rightarrow 0$, $2n\rightarrow n$, $2n+1\rightarrow
3n+2$ (for example, $P_5(z) = 5 + 8z + 4z^2 + 2z^3 + z^4$). In this article, we
prove that all zeros of $P_N$ (which we call Collatz zeros) lie in an annulus
centered at the origin, with outer radius 2 and inner radius a function of the
largest odd iterate of $N$. Moreover, using an extension of the
Enestr\"om-Kakeya Theorem, we prove that $|z| = 2$ for a root of $P_N$ if and
only if the Collatz trajectory of $N$ has a certain form; as a corollary, the
set of $N$ for which our upper bound is an equality is sparse in $\mathbb{N}$.
Inspired by these results, we close with some questions for further study.
