  In this text, we study factorizations of polynomials over the tropical
hyperfield and the sign hyperfield, which we call `tropical polynomials' and
`sign polynomials', respectively. We classify all irreducible polynomials in
either case. We show that tropical polynomials factor uniquely into irreducible
factors, but that unique factorization fails for sign polyomials. We describe
division algorithms for tropical and sign polynomials by linear terms that
correspond to roots of the polynomials.
