14: Polynomial Functions
- Page ID
- 174372
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 14.1: Power Functions
- This page provides an overview of power functions defined as \(f(x) = ax^n\) where \(a \neq 0\) and \(n\) is a positive integer, highlighting their distinction from exponential functions. It discusses even and odd functions, their symmetry, and end behavior, framed by relevant theorems.
- 14.2: The Basics of Polynomial Functions
- This page covers polynomial functions, detailing their general form, degree, and leading coefficient. It explores important characteristics like end behavior and turning points, incorporating the Leading Coefficient Test to predict end behavior. Theorems on polynomial behavior are discussed, along with restrictions on turning points.
- 14.3: The Factor and Remainder Theorems
- This page explains the Factor and Remainder Theorems for polynomials, defining zeros and detailing the relationship between factors and division. It states that the remainder of dividing \(f(x)\) by \(x-c\) equals \(f(c)\), and links zeros to factors. It warns about common errors in application.
- 14.4: Complex Zeros and the Fundamental Theorem of Algebra
- This page covers essential concepts regarding polynomials and their zeros, including definitions, the Fundamental Theorem of Algebra, and the Complete Factorization Theorem. It highlights that nonconstant polynomials have at least one complex zero, and complex zeros occur as conjugate pairs in polynomials with real coefficients.
- 14.5: Graphing Polynomial Functions
- This page provides an overview of polynomial functions, covering key concepts such as the definition of zeros and their multiplicities, the impact of degree on graph characteristics and end behavior, and strategies for sketching polynomial graphs. It emphasizes continuity, smoothness, and the significance of real zeros and their multiplicities, with practical examples illustrating these concepts.
- 14.6: Applications Involving Polynomial Functions
- This page covers polynomial functions and their real-world applications, defining key concepts such as polynomial models, local/global extrema, and continuity. It emphasizes the use of technology to locate extrema and includes theorems on extremum and turning points. Graphical methods for solving polynomial equations are explored, with examples on maximizing volume, revenue, and profit, while also addressing break-even points.
- 14.7: Pascal's Triangle
- This page covers factorials, binomial coefficients, and Pascal's Triangle, defining the factorial and introducing the binomial coefficient \(\binom{n}{k}\). It explains Pascal's Triangle and key theorems including Pascal's Rule and the Binomial Theorem, vital for combinatorial and polynomial tasks.
- 14.8: The Binomial Theorem
- This page covers the Binomial Theorem, detailing its definitions, components like binomial coefficients and Pascal's Triangle, and providing a proof. It explains the expansion of \((a+b)^n\), illustrating the process with examples and the significance of indexing. Key applications include expanding specific binomials and calculating coefficients.