Rational and Algebraic Expressions and Functions

Abstract

As a natural continuation of the study of polynomials, in this chapter we introduce and discuss rational and algebraic expressions in a wide variety of settings. One of the main objectives of this chapter is to present the partial fraction decomposition in complete details; this is accompanied by a few Olympiad level problems. Asymptotes, briefly alluded to in treating hyperbolas in Section 8.4, are fully and rigorously developed here. Another main objective of this chapter is to extend the AM–GM inequality (Sections 5.4, 7.5) to the multivariate harmonic–geometric–arithmetic–quadratic mean inequalities. The AM–GM inequality along with its extensions is a cornerstone of analysis. It has a beautiful geometry which was known to the ancient Greeks, and it appears in a myriad problems such as multivariate extremal problems, factorization problems, etc. Among the literally hundreds of mathematical contest problems involving these means, we chose a representative sample to demonstrate the principal methods. The lesser known permutation (arrangement) inequality is also introduced here pointing out that it implies all the other classical inequalities such as the AM–GM, Cauchy–Schwarz (Sections 5.3, 6.7), and Chebyshev (Section 6.7) inequalities. Finally, we give a detailed (and somewhat more advanced) account on the greatest integer function along with some of Ramanujan’s formulas, and the Hermite identity.