Math 102/Chapter 5

Chapter 5: Polynomial and Rational Functions

Chapter 5: Polynomial and Rational Functions

• Polynomial functions and models
• Properties of rational functions
• Graphs of rational functions
• Polynomial and rational inequalities
• Real zeros of polynomials

Section 5.1: Polynomial functions and models

Polynomial function definition

Identifying polynomial functions

Graphing polynomial function using transformations

Finding polynomial function from its zeros

Identifying zeros and their multiplicities

Graphing polynomial using x-intercepts

Identifying the graph of a polynomial function

Identifying graph of polynomial function

Graph of polynomial function: summary

How to analyze graph of polynomial function

Analyzing graph of polynomial function

How to use graphing utility to assist in analyzing graph of polynomial

Steps for using graphing utility to analyze polynomial

Cubic function of best fit

Section 5.2: Properties of rational functions

Finding domain of rational function

Graphing y = 1/x2

Using transformations to graph rational functions

Finding vertical asymptotes of rational function

Finding vertical asymptotes

Finding horizontal asymptotes

Finding horizontal or oblique asymptotes

Finding horizontal or oblique asymptotes

Finding horizontal or oblique asymptotes

Section 5.3: Graph of rational function

How to analyze the graph of a rational function

Step by step solution: how to analyze graph of rational function

Analyzing graph of rational function

Analyzing graph of rational function

Analyzing graph of rational function

Analyzing graph of rational function with hole

Constructing rational function from graph

Finding least cost of a can

Section 5.4: Polynomial and rational inequalities

Solving polynomial inequality using graph

Solving polynomial inequality algebraically

Solving rational inequality using its graph

How to solve a rational inequality algebraically

Section 5.5: Real zeros of polynomial function

Remainder theorem

Factor theorem

Division algorithm for polynomials

Using remainder theorem

Using factor theorem

Rational zeros theorem

Listing potential rational zeros

How to find real zeros of polynomial function

Finding zeros of polynomial function

Steps: finding real zeros of polynomial function

Solving polynomial equation

Using intermediate value theorem to find bounds on zeros

using intermediate value theorem to locate a real zero

Approximating a real zero of a polynomial function

Section 5.6: Complex zeros, fundamental theorem of algebra

Fundamental theorem of algebra

Conjugate pairs theorem

Corollary

Using conjugate pairs theorem

Finding polynomial function whose zeros are given

Finding complex zeros of polynomial function

Chapter 5 Objectives

Section 5.1:

• Identify polynomial functions and their degree
• Graph polynomial functions using transformations
• Identify the real zeros of a polynomial function and their multiplicity
• Analyze the graph of a polynomial function
• Build cubic models from data

Section 5.2:

• Find the domain of a rational function
• Find the vertical asymptotes of a rational function
• Find the horizontal or oblique asymptote of a rational function

Section 5.3:

• Analyze the graph of a rational function
• Solve applied prob.ems involving rational functions

Section 5.4:

• Solve polynomial inequalities
• Solve rational inequalities

Section 5.5:

• Use the remainder and factor theorems
• Use the rational zeros theorem to list the potential rational zeros of a polynomial function
• Find the real zeros of a polynomial funciton
• Solve opoynoial equations
• Use the Theorem for Bounds on Zeros
• Use intermediate value theorem

Section 5.6:

• Use the conjugate pairs theorem
• Find a polynomial function with specified zeros
• Find the complex zeros of a polynomial