2: 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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- 2.0: Introduction to Polynomial and Rational Functions
- This chapter is dedicated to polynomial and rational functions. We explore their graphs, including the behavior of these curves near their zeros. Vertical, horizontal, and slant asymptotes are detailed when discussing rational functions. Additionally, we spend quite some time working with the Remainder and Factor Theorems in an effort to find all roots (both real- and complex-valued) of polynomials. Synthetic division and the Rational Zeros Theorem are introduced to help with this process.
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- 2.1: Complex Numbers
- This section introduces complex numbers, covering their standard form \(a + bi\), where \(i\) is the imaginary unit. It explains operations with complex numbers, including addition, subtraction, multiplication, division, and finding complex conjugates. The section also introduces the complex plane, where complex numbers are represented graphically, and discusses their magnitude and applications. Examples illustrate these concepts in practice.
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- 2.2: Quadratic Functions
- This section covers quadratic functions, focusing on their general and standard (vertex) forms. It explains how to find and interpret key features such as the vertex, axis of symmetry, and zeros. It also demonstrates graphing parabolas and solving quadratic equations using the Quadratic Formula, completing the square, and factoring. Advanced topics include solving absolute value equations involving quadratic expressions and graphing the absolute value of a quadratic function.
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- 2.3: Power Functions and Polynomial Functions
- This section discusses power and polynomial functions, focusing on their definitions, properties, and graphs. It explains the general form of polynomial functions, the significance of the leading term and degree, and how these influence end behavior. Examples demonstrate graphing polynomial functions and analyzing their key features, including turning points and continuity.
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- 2.4: Graphs of Polynomial Functions
- This section explores the graphs of polynomial functions, focusing on key characteristics such as end behavior, intercepts, and turning points. It discusses how the degree and leading coefficient influence the graph's shape and analyzes multiplicity's role in determining the behavior of the graph at roots. Examples illustrate how to sketch and interpret polynomial graphs effectively.
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- 2.5: Dividing Polynomials
- This section covers methods for dividing polynomials, including long division and synthetic division. It explains how to use these techniques to divide a polynomial by a linear or higher-degree polynomial, interpret the results, and find remainders. Examples illustrate each method step-by-step, helping to solve polynomial division problems efficiently.
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- 2.6: Zeros of Polynomial Functions
- This section explores zeros of polynomial functions, focusing on identifying roots using the Rational Zeros Theorem, the Fundamental Theorem of Algebra, the Complete Factorization Theorem, and the Conjugate Pairs Theorem. It examines how multiplicity impacts the graph of a polynomial at its roots and provides techniques for solving higher-degree polynomial equations through factoring and other algebraic methods.
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- 2.7: Rational Functions
- This section introduces rational functions, exploring their key features such as domain, vertical and horizontal or slant asymptotes, and intercepts. It discusses how to analyze and graph rational functions, focusing on behavior near asymptotes and at infinity. Examples demonstrate the steps to identify these features and interpret their implications for the function's graph.
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- 2.8: Revisiting Inverses
- This section discusses the inverses of polynomial and rational functions, focusing on determining whether a function has an inverse, finding the inverse algebraically, and understanding the domain and range restrictions necessary for invertibility. It provides examples to demonstrate these concepts and explores graphical relationships between functions and their inverses, including symmetry about the line \(y = x\).
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