  For a positive real number $\alpha$, let $\mathbb{N}_0[\alpha,\alpha^{-1}]$
be the semiring of all real numbers $f(\alpha)$ for $f(x)$ lying in
$\mathbb{N}_0[x,x^{-1}]$, which is the semiring of all Laurent polynomials over
the set of nonnegative integers $\mathbb{N}_0$. In this paper, we study various
factorization properties of the additive structure of $\mathbb{N}_0[\alpha,
\alpha^{-1}]$. We characterize when $\mathbb{N}_0[\alpha, \alpha^{-1}]$ is
atomic. Then we characterize when $\mathbb{N}_0[\alpha, \alpha^{-1}]$ satisfies
the ascending chain condition on principal ideals in terms of certain
well-studied factorization properties. Finally, we characterize when
$\mathbb{N}_0[\alpha, \alpha^{-1}]$ satisfies the unique factorization property
and show that, when this is not the case, $\mathbb{N}_0[\alpha, \alpha^{-1}]$
has infinite elasticity.
