A remark on roots of polynomials with positive coefficients (Q833081)

There are several ways to prove that a complex number \(\alpha\) is a root of a polynomial with positive rational coefficients if and only if \(\alpha\) is an algebraic number over \(\mathbb{Q}\) without non-negative real conjugates. In particular, this was proved by the reviewer [Manuscr. Math. 123, 353--356 (2007; Zbl 1172.11036)], but also follows from some earlier results too. In this note, the author observes that this result can be proved using the following result of Klamkin (1952) and Handelman (1985): if \(f(x) \in \mathbb{R}[x]\) is a polynomial having no non-negative roots then there is an \(m \in \mathbb{N}\) such that \((1+x)^m f(x)\) has only positive coefficients.