I hope to convince you that questions about factorizations of polynomials with non-negative real coefficients---not necessarily into irreducibles and possibly with conditions on the degrees of the factors---are worth studying by discussing several examples that have interesting answers. The question in the title arises from an equivalent formulation that asks what we can say about a finite collection of dice from knowing the probability distribution of the totals that arise when the dice are rolled. No knowledge of the game of craps will be required, but you may enjoy preparing by streaming this classic scene
from the movie "A Bronx Tale" set in the 1960s in the Belmont neighborhood next to Fordham University where I teach