  In this paper, we study factorizations in the additive monoids of positive
algebraic valuations $\mathbb{N}_0[\alpha]$ of the semiring of polynomials
$\mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F.
Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic
if every non-invertible element factors into irreducibles. We begin by
determining when $\mathbb{N}_0[\alpha]$ is atomic, and we give an explicit
description of its set of irreducibles. An atomic monoid is a finite
factorization monoid (FFM) if every element has only finitely many
factorizations (up to order and associates), and it is a bounded factorization
monoid (BFM) if for every element there is a bound for the number of
irreducibles (counting repetitions) in each of its factorizations. We show
that, for the monoid $\mathbb{N}_0[\alpha]$, the property of being a BFM and
the property of being an FFM are equivalent to the ascending chain condition on
principal ideals (ACCP). Finally, we give various characterizations for
$\mathbb{N}_0[\alpha]$ to be a unique factorization monoid (UFM), two of them
in terms of the minimal polynomial of $\alpha$. The properties of being
finitely generated, half-factorial, and length-factorial are also investigated
along the way.
