
# Chapter 3: Polynomial and Rational Functions


In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications.

• Section 3.1: Complex Numbers
After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.
• Section 3.2: Quadratic Functions
In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.
• Section 3.3: Power Functions and Polynomial Functions
Suppose a certain species of bird thrives on a small island. The population can be estimated using a polynomial function. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.
• Section 3.4: Graphs of Polynomial Functions
The revenue in millions of dollars for a fictional cable company can be modeled by the polynomial function  From the model one may be interested in which intervals the revenue for the company increase or decreases? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.
• Section 3.5: Dividing Polynomials
We are familiar with the long division algorithm for ordinary arithmetic. We begin by dividing into the digits of the dividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value position,. Division of polynomials that contain more than one term has similarities to long division of whole numbers. We can write a polynomial dividend as the product of the divisor and the quotient added to the remainder.
• Section 3.6: Zeros of Polynomial Functions
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by $$x–k$$, the remainder may be found quickly by evaluating the polynomial function at $$k$$, that is, $$f(k)$$.
• Section 3.7: Rational Functions
In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.
• Section 3.8: Inverses and Radical Functions
In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process.
• Section 3.9: Polynomial and Rational Inequalities