4: Polynomial and Rational Functions
In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications.
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- 4.1: 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.
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- 4.2: 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.
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- 4.3: 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.
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- 4.4: 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, \(f(k)\).
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- 4.5: Simplifying Rational Expressions
- The goal of this section is to learn how to reduce a rational expression to “lowest terms.” Of course, that means that we will have to understand what is meant by the phrase “lowest terms.” With that thought in mind, we begin with a discussion of the greatest common divisor of a pair of integers.
Contributors and Attributions
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Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .