From the text: “In a recent note [Am. Math. Mon. 111, No. 6, 525–526 (2004; Zbl 1080.26507)] X.-S. Wang gave a simple proof of a generalization of Descartes’s rule of signs. We give another simple proof of Wang’s result.
Theorem. Let \(p(x)=a_0x^{b_0}+a_1x^{b_1}+\ldots+a_nx^{b_n}\) be a function with nonzero real coefficients \(a_0,a_1,\ldots,a_n\) and real exponents \(b_0,b_1,\ldots,b_n\) satisfying \(b_0>b_1>\ldots>b_n\). Then \(p\) cannot have more positive roots (even counted with multiplicity) than the number of sign changes in the sequence \(a_0,\ldots,a_n\).”
The difference in proving this result is clearly explained and a short, but interesting historical origin of this theorem and its proof is also given. It was first stated by Descartes without proof in 1637, the first proofs were given by J. A. Segner (1728,1756) (vgl. W. Jentsch [Die Bedeutung Johann Andreas Segners in der Geschichte der Mathematik. In: Kaiser, Wolfram (ed.) et al., Johann Andreas Segner (1704–1777) und seine Zeit: Hallesches Segner-Symposium 1977. Wiss. Beitr., Martin-Luther-Univ. Halle-Wittenberg 1977, No. 36 (T20), 152–158 (1977); A. Kleinert, Johann Andreas (von) Segner (1704–1777). Halle: Univ. Halle-Wittenberg, Fachbereich Mathematik und Informatik, Reports on Didactics and History of Mathematics 19, 15–20 (2002)]).