Abstract
This chapter revisits, for real polynomials and departing from the fundamental theorem of Algebra, some classical theorems of Calculus. As applications of them, we shall prove Newton’s inequalities, which generalizes the classical inequality between the arithmetic and geometric means of n positive real numbers, and Descartes’ rule, which relates the number of positive roots of a real polynomial with the number of changes of sign in the sequence of its nonzero coefficients.
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Notes
- 1.
Bernhard Bolzano , German mathematician of the nineteenth century.
- 2.
René Descartes , French mathematician, philosopher and scientist of the seventeenth century. Descartes’ legacy to Mathematics and science is a huge one, and came mainly from his landmarking book Discours de la Méthode (Discourse on the Method) and its three corresponding appendices. This book marks a turning point on the way of doing science, for, along it, Descartes strongly rejected the scholastic tradition of using speculation, instead of deduction, as the central strategy for the investigation of natural phenomena. On the other hand, its appendix La Géométrie layed down the foundations of Analytic Geometry, and nowadays every student is acquainted with cartesian coordinate systems.
References
T. Apostol, Calculus, Vol. 1 (Wiley, New York, 1967)
A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)
A.G. Kurosch, Curso de Algebra Superior (in Spanish) (MIR, Moscow, 1968)
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Caminha Muniz Neto, A. (2018). Polynomials Over \(\boldsymbol {\mathbb R}\). In: An Excursion through Elementary Mathematics, Volume III. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77977-5_17
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DOI: https://doi.org/10.1007/978-3-319-77977-5_17
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