Unit 5: Polynomial Functions
Polynomial functions include quadratic functions, which may be familiar to you from past math courses. In this unit, we will continue the pattern of defining key characteristics of polynomial functions such as the domain and range using equations and graphs. In addition to quadratic functions, the family of polynomial functions includes functions with higher degree exponents. As the degree of a polynomial increases, the algebra required to define its key characteristics such as intercepts and inflection points becomes more and more complicated. You will learn techniques to describe the behaviors of polynomial functions that help us understand them without performing complex algebra.
Completing this unit should take you approximately 4 hours.
Upon successful completion of this unit, you will be able to:
- identify the properties of power, quadratic, and polynomial functions given a graph or an equation including end behavior, domain and range, and local behaviors by using algebraic operations, interval notation, and words;
- relate elements of the standard form of a quadratic function with corresponding transformations to its graph;
- construct the equation of a power, quadratic, or polynomial function given its graph;
- apply the intermediate value theorem to determine the approximate location of a zero of a polynomial;
- evaluate polynomial functions using the remainder and factor theorems;
- apply the rational zeros theorem, the fundamental theorem of algebra, and the linear factorization theorem to identify zeros of polynomials; and
- confirm numbers of positive and negative roots of polynomials using Descartes' rule of signs.
5.1: Quadratic Functions
We will continue our study of functions by exploring the characteristics of quadratic functions. You may have solved quadratic equations in the past, and now we will bring together the quadratic equation and the quadratic function. We will explore the graph of a quadratic function and use some of the same techniques for solving quadratic equations to find special points on the function. You will learn how to identify the vertex, axis of symmetry, and intercepts of the graph of a quadratic function and how to calculate their value given the equation of a quadratic function.
The domain and range of a function are integral to its definition. In this section, you will learn how to use algebraic techniques to define a function's domain and range given its equation.
You will use the ideas in this section to define key characteristics of polynomial and rational graphs. These ideas continue into first-year calculus and help us analyze behaviors of functions and trends in general.
5.2: Power and Polynomial Functions
In this section, you will learn how to identify a power function and use interval notation to express its long-run behavior. If you need a refresher on how to use interval notation, now is a good time to review.
Now, you will learn how to identify a polynomial function and what makes them different from a power function. You will also be able to define the key characteristics of a polynomial function, such as the degree, leading coefficient, end behavior, intercepts, and turning points.
5.3: Graphs of Polynomial Functions
This section will dig deeper into the relationship between the graph of a polynomial function and its equation. You will see how to use the factors of a polynomial function to determine where the x-intercepts are, and you will also learn about the multiplicity of a zero (x-intercept) and how to find it.
In this section, we will bring all we know about polynomial functions and use it to sketch a graph given an equation. You will also learn about the intermediate value theorem and how we can use it to analyze behaviors when we don't know exactly where the zeros of a polynomial are.
5.4: Dividing Polynomials
This section will give you the algebraic skills to find zeros of polynomials without the aid of a graphing utility. We will use the skills learned here in the coming sections on finding zeros of polynomials and rational functions. If it has been a while, you may need to recall how to use long division to divide integers.
Synthetic division is another algebraic method for finding roots (x-intercepts) of polynomials. Some people prefer this method because it is tidier than long division. With synthetic division, it is important to understand how to interpret your results.
5.5: Finding Roots of Polynomial Functions
In this section, we will apply polynomial division techniques to analyze and evaluate polynomials. You will be able to evaluate a polynomial function for a given value using the remainder theorem and the factor theorem. These two techniques work well when the roots of a polynomial are integers. We need to use the rational zeros theorem when we have rational roots. This technique also uses polynomial division but will yield zeros that are rational numbers.
After you finish this section, you will be able to find the complex zeros of a polynomial function. This is the last stage in finding zeros of polynomial functions.