Polynomial and Rational Functions

This chapter widens the supply of functions past the linear ones of Chapter 2, and it starts by widening the numbers themselves: the square roots of negative numbers that quadratic equations keep producing become a number system of their own. From there it works through the polynomial family — quadratics and the optimization problems they answer, the power functions that set a polynomial’s end behavior, what a polynomial’s graph does at each of its zeros, and the division and factoring theorems that locate those zeros in the first place. It closes with the functions built by dividing and inverting polynomials, and with the variation models that describe one quantity growing as a power of another.

Sections

• Complex Numbers — expressing square roots of negative numbers as multiples of $i$, plotting complex numbers on the complex plane, and adding, subtracting, multiplying, and dividing them.
• Quadratic Functions — recognizing the characteristics of parabolas, relating a parabola’s graph to its general and standard forms, and finding a quadratic function’s minimum or maximum value and the problems it answers.
• Power Functions and Polynomial Functions — identifying power functions and their end behavior, recognizing polynomial functions, and reading off the degree and leading coefficient that govern how a graph behaves far from the origin.
• Graphs of Polynomial Functions — the characteristics of polynomial graphs, finding zeros by factoring, zeros and their multiplicities, end behavior, the relationship between degree and turning points, sketching a graph from those pieces, and the Intermediate Value Theorem.
• Dividing Polynomials — dividing polynomials by long division and by synthetic division, and using polynomial division to solve area and volume problems.
• Zeros of Polynomial Functions — the Remainder and Factor Theorems, the Rational Zero Theorem, finding a polynomial’s zeros, building a polynomial from given zeros with the Linear Factorization Theorem, Descartes’ Rule of Signs, and real-world applications.
• Rational Functions — arrow notation, the domains of rational functions, vertical and horizontal asymptotes, graphing rational functions, and the applied problems they model.
• Inverses and Radical Functions — finding the inverse of a polynomial function, and restricting the domain so that an inverse exists.
• Modeling Using Variation — solving direct, inverse, and joint variation problems.

This chapter is adapted from Precalculus 2e, Chapter 3: Polynomial and Rational Functions by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Each section page records its own adaptations.