A simple method of finding limits for the absolute values of the zeros of polynomials will be given in this paper. The radius of a circle about the origin in the complex plane will be found from the coefficient of the polynomial such that all the zeros will lie on or within this circle. For certain polynomials a second circle will be found such that the zeros will lie on or outside this circle. In finding these circles the following well-known theorem of Rouché is used. Rouché's Theorem is very useful. By means of it the fundamental theorem of algebra and other important results may be established. The proof of Rouché's Theorem is ordinarily based on residue theory and may be found in most books on the theory of functions of a complex variable. Rouché, however, used series expansions to prove his theorem.
Proceedings of the Iowa Academy of Science
©1951 Iowa Academy of Science, Inc.
Stoner, Wm. J.
"Theorem on the Zeros of Polynomials,"
Proceedings of the Iowa Academy of Science, 58(1), 311-312.
Available at: https://scholarworks.uni.edu/pias/vol58/iss1/37