# 3: Polynomial Functions

- Page ID
- 3995

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

- 3.1: Graphs of Polynomials
- Three of the families of functions studied thus far: constant, linear and quadratic, belong to a much larger group of functions called polynomials. We begin our formal study of general polynomials with a definition and some examples.

- 3.2: The Factor Theorem and the Remainder Theorem
- Suppose we wish to find the zeros of an arbitrary polynomial. Even though we could use the 'Zero' command to find decimal approximations for these, we seek a method to find the remaining zeros exactly. The point of this section is to generalize the technique applied here. First up is a friendly reminder of what we can expect when we divide polynomials.

- 3.3: Real Zeros of Polynomials
- This section presents results which will help us determine good candidates to test using synthetic division. There are two approaches to the topic of finding the real zeros of a polynomial. The first approach (which is gaining popularity) is to use a little bit of Mathematics followed by a good use of technology like graphing calculators. The second approach (for purists) makes good use of mathematical machinery (theorems) only. For completeness, we include both approaches.

- 3.4: Complex Zeros and the Fundamental Theorem of Algebra
- Previously, we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and look for the non-real zeros as well. The requires introducing the imaginary unit, i, that while not a real number, plays along well with real numbers, and acts very much like any other radical expression

- 3.E: Polynomial Functions (Exercises)
- These are homework exercises to accompany Chapter 3 of Stitz and Zeager's "Pre-Calculus" Textmap.

## Contributors

- Carl Stitz, Ph.D. (Lakeland Community College) and Jeff Zeager, Ph.D. (Lorain County Community College)