COLLEGE ALGEBRA - Wichita

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COLLEGE ALGEBRAPaul Dawkins

College AlgebraTable of ContentsPreface . iiiOutline . ivPreliminaries . 1Introduction . 1Integer Exponents . 2Rational Exponents . 9Real Exponents .15Radicals .16Polynomials.25Factoring Polynomials .31Rational Expressions .41Complex Numbers .52Solving Equations and Inequalities. 58Introduction .58Solutions and Solution Sets .59Linear Equations .63Application of Linear Equations .71Equations With More Than One Variable .81Quadratic Equations – Part I .85Quadratic Equations – Part II .93Solving Quadratic Equations : A Summary .104Application of Quadratic Equations .107Equations Reducible to Quadratic Form .111Equations with Radicals .116Linear Inequalities.122Polynomial Inequalities .129Rational Inequalities .135Absolute Value Equations .140Absolute Value Inequalities .147Graphing and Functions . 152Introduction .152Graphing .153Lines .159Circles .169The Definition of a Function .175Graphing Functions .188Combining Functions .192Inverse Functions .199Common Graphs . 206Introduction .206Lines, Circles and Piecewise Functions .207Parabolas .208Ellipses .218Hyperbolas .222Miscellaneous Functions .226Transformations .229Symmetry .235Rational Functions .240Polynomial Functions . 246Introduction .246Dividing Polynomials .247Zeroes/Roots of Polynomials .252 2007 Paul Dawkinsihttp://tutorial.math.lamar.edu/terms.aspx

College AlgebraGraphing Polynomials .257Finding Zeroes of Polynomials .265Partial Fractions .273Exponential and Logarithm Functions . 281Introduction .281Exponential Functions .282Logarithm Functions .287Solving Exponential Equations .297Solving Logarithm Equations .304Applications .310Systems of Equations. 317Introduction .317Linear Systems with Two Variables .318Linear Systems with Three Variables .326Augmented Matrices .330More on the Augmented Matrix .339Non-Linear Systems .345 2007 Paul Dawkinsiihttp://tutorial.math.lamar.edu/terms.aspx

College AlgebraPrefaceHere are my online notes for my Algebra course that I teach here at Lamar University, although Ihave to admit that it’s been years since I last taught this course. At this point in my career Imostly teach Calculus and Differential Equations.Despite the fact that these are my “class notes”, they should be accessible to anyone wanting tolearn Algebra or needing a refresher for algebra. I’ve tried to make the notes as self contained aspossible and do not reference any book. However, they do assume that you’ve had someexposure to the basics of algebra at some point prior to this. While there is some review ofexponents, factoring and graphing it is assumed that not a lot of review will be needed to remindyou how these topics work.Here are a couple of warnings to my students who may be here to get a copy of what happened ona day that you missed.1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learnalgebra I have included some material that I do not usually have time to cover in classand because this changes from semester to semester it is not noted here. You will need tofind one of your fellow class mates to see if there is something in these notes that wasn’tcovered in class.2. Because I want these notes to provide some more examples for you to read through, Idon’t always work the same problems in class as those given in the notes. Likewise, evenif I do work some of the problems in here I may work fewer problems in class than arepresented here.3. Sometimes questions in class will lead down paths that are not covered here. I try toanticipate as many of the questions as possible in writing these up, but the reality is that Ican’t anticipate all the questions. Sometimes a very good question gets asked in classthat leads to insights that I’ve not included here. You should always talk to someone whowas in class on the day you missed and compare these notes to their notes and see whatthe differences are.4. This is somewhat related to the previous three items, but is important enough to merit itsown item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!Using these notes as a substitute for class is liable to get you in trouble. As already notednot everything in these notes is covered in class and often material or insights not in thesenotes is covered in class. 2007 Paul x

College AlgebraOutlineHere is a listing of all the material that is currently available in these notes.PreliminariesInteger Exponents – In this section we will start looking at exponents and theirproperties.Rational Exponents – We will define rational exponents in this section and extend theproperties from the previous section to rational exponents.Real Exponents – This is a short acknowledgment that the exponent properties from theprevious two sections will hold for any real exponent.Radicals – Here we will define radical notation and relate radicals to rational exponents.We will also give the properties of radicals.Polynomials – We will introduce the basics of polynomials in this section includingadding, subtracting and multiplying polynomials.Factoring Polynomials – This is the most important section of all the preliminaries.Factoring polynomials will appear in pretty much every chapter in this course. Withoutthe ability to factor polynomials you will be unable to complete this course.Rational Expressions – In this section we will define rational expressions and discussadding, subtracting, multiplying and dividing them.Complex Numbers – Here is a very quick primer on complex numbers and how tomanipulate them.Solving Equations and InequalitiesSolutions and Solution Sets – We introduce some of the basic notation and ideasinvolved in solving in this section.Linear Equations – In this section we will solve linear equations, including equationswith rational expressions.Applications of Linear Equations – We will take a quick look at applications of linearequations in this section.Equations With More Than One Variable – Here we will look at solving equationswith more than one variable in them.Quadratic Equations, Part I – In this section we will start looking at solving quadraticequations. We will look at factoring and the square root property in this section.Quadratic Equations, Part II – We will finish up solving quadratic equations in thissection. We will look at completing the square and quadratic formula in this section.Quadratic Equations : A Summary – We’ll give a procedure for determining whichmethod to use in solving quadratic equations in this section. We will also take a quicklook at the discriminant.Applications of Quadratic Equations – Here we will revisit some of the applicationswe saw in the linear application section, only this time they will involve solving aquadratic equation.Equations Reducible to Quadratic Form – In this section we will solve equations thatcan be reduced to quadratic in form.Equations with Radicals – Here we will solve equations with square roots in them. 2007 Paul Dawkinsivhttp://tutorial.math.lamar.edu/terms.aspx

College AlgebraLinear Inequalities – We will start solving inequalities in this section by looking atlinear inequalities.Polynomial Inequalities – In this section we will look at solving inequalities that containpolynomials.Rational Inequalities – Here we will solve inequalities involving rational expressions.Absolute Value Equations – We will officially define absolute value in this section andsolve equations that contain absolute value.Absolute Value Inequalities – We will solve inequalities that involve absolute value inthis section.Graphing and FunctionsGraphing – In this section we will introduce the Cartesian coordinate system and most ofthe basics of graphing equations.Lines – Here we will review the main ideas from the study of lines including slope andthe special forms of the equation of a line.Circles – We will look at the equation of a circle and graphing circles in this section.The Definition of a Function – We will discuss the definition of a function in thissection. We will also introduce the idea of function evaluation.Graphing Functions – In this section we will look at the basics of graphing functions.We will also graph some piecewise functions in this section.Combining functions – Here we will look at basic arithmetic involving functions as wellas function composition.Inverse Functions – We will define and find inverse functions in this section.Common GraphsLines, Circles and Piecewise Functions – This section is here only to acknowledge thatwe’ve already talked about graphing these in a previous chapter.Parabolas – We’ll be graphing parabolas in this section.Ellipses – In this section we will graph ellipses.Hyperbolas – Here we will be graphing hyperbolas.Miscellaneous Functions – In this section we will graph a couple of common functionsthat don’t really take all that much work to so. We’ll be looking at the constant function,square root, absolute value and a simple cubic function.Transformations – We will be looking at shifts and reflections of graphs in this section.Collectively these are often called transformations.Symmetry – We will briefly discuss the topic of symmetry in this section.Rational Functions – In this section we will graph some rational functions. We will alsobe taking a look at vertical and horizontal asymptotes.Polynomial FunctionsDividing Polynomials – We’ll review some of the basics of dividing polynomials in thissection.Zeroes/Roots of Polynomials – In this section we’ll define just what zeroes/roots ofpolynomials are and give some of the more important facts concerning them.Graphing Polynomials – Here we will give a process that will allow us to get a roughsketch of some polynomials.Finding Zeroes of Polynomials – We’ll look at a process that will allow us to find someof the zeroes of a polynomial and in special cases all of the zeroes.Partial Fractions – In this section we will take a look at the process of partial fractionsand finding the partial fraction decomposition of a rational expression. 2007 Paul Dawkinsvhttp://tutorial.math.lamar.edu/terms.aspx

College AlgebraExponential and Logarithm FunctionsExponential Functions – In this section we will introduce exponential functions. Wewill be taking a look at some of the properties of exponential functions.Logarithm Functions – Here we will introduce logarithm functions. We be looking athow to evaluate logarithms as well as the properties of logarithms.Solving Exponential Equations – We will be solving equations that containexponentials in this section.Solving Logarithm Equations – Here we will solve equations that contain logarithms.Applications – In this section we will look at a couple of applications of exponentialfunctions and an application of logarithms.Systems of EquationsLinear Systems with Two Variables – In this section we will use systems of twoequations and two variables to introduce two of the main methods for solving systems ofequations.Linear Systems with Three Variables – Here we will work a quick example to showhow to use the methods to solve systems of three equations with three variables.Augmented Matrices – We will look at the third main method for solving systems inthis section. We will look at systems of two equations and systems of three equations.More on the Augmented Matrix – In this section we will take a look at some specialcases to the solutions to systems and how to ident

exposure to the basics of algebra at some point prior to this. While there is some review of exponents, factoring and graphing it is

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