CHAPTER ELEVEN - ROOTS OF POLYNOMIALS

Abstract

This chapter discusses the roots of polynomials. It presents answers for the following questions for a polynomial equation of degree n: (1) how many roots does a polynomial have in the field of complex numbers?, (2) how many of the roots of a polynomial are real numbers?, (3) if the coefficients of a polynomial are integers, how many of the roots are rational numbers?, and (4) is there a relationship between the roots and factors of a polynomial? The chapter discusses some information concerning the number of positive real roots and the number of negative real roots of such polynomials. If the terms of a polynomial with real coefficients are written in descending order, then a variation in sign occurs whenever two successive terms have opposite signs. In determining the number of variations of sign, terms with zero coefficients are ignored. The French mathematician Rene Descartes provided the foundations of analytic geometry and also gave a theorem that relates the nature of the real roots of polynomials to the variations in sign. The chapter discusses Descartes' rule of signs.