Paul's Online Math Notes
Algebra (Notes)   [Notes] [Practice Problems] [Assignment Problems]


On August 21 I am planning to perform a major update to the site. I can't give a specific time in which the update will happen other than probably sometime between 6:30 a.m. and 8:00 a.m. (Central Time, USA). There is a very small chance that a prior commitment will interfere with this and if so the update will be rescheduled for a later date.

I have spent the better part of the last year or so rebuilding the site from the ground up and the result should (hopefully) lead to quicker load times for the pages and for a better experience on mobile platforms. For the most part the update should be seamless for you with a couple of potential exceptions. I have tried to set things up so that there should be next to no down time on the site. However, if you are the site right as the update happens there is a small possibility that you will get a "server not found" type of error for a few seconds before the new site starts being served. In addition, the first couple of pages will take some time to load as the site comes online. Page load time should decrease significantly once things get up and running however.

August 7, 2018

Algebra - Notes
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  Next Section Preliminaries (Introduction)
College Algebra (Math 1314)

Here are my online notes for my Algebra course that I teach here at Lamar University, although I have to admit that it’s been years since I last taught this course.  At this point in my career I mostly teach Calculus and Differential Equations. 


Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn Algebra or needing a refresher for algebra.  I’ve tried to make the notes as self contained as possible and do not reference any book.  However, they do assume that you’ve had some exposure to the basics of algebra at some point prior to this.  While there is some review of exponents, factoring and graphing it is assumed that not a lot of review will be needed to remind you how these topics work.


Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. 


  1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn algebra I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here.  You will need to find one of your fellow class mates to see if there is something in these notes that wasn’t covered in class.

  2. Because I want these notes to provide some more examples for you to read through, I don’t always work the same problems in class as those given in the notes.  Likewise, even if I do work some of the problems in here I may work fewer problems in class than are presented here.

  3. Sometimes questions in class will lead down paths that are not covered here.  I try to anticipate as many of the questions as possible in writing these up, but the reality is that I can’t anticipate all the questions.  Sometimes a very good question gets asked in class that leads to insights that I’ve not included here.  You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.

  4. This is somewhat related to the previous three items, but is important enough to merit its own 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 noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.


Here is a listing of all the material that is currently available in these notes.



Integer Exponents  In this section we will start looking at exponents and their properties.

Rational Exponents  We will define rational exponents in this section and extend the properties from the previous section to rational exponents.

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 including adding, 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.  Without the ability to factor polynomials you will be unable to complete this course.

Rational Expressions  In this section we will define rational expressions and discuss adding, subtracting, multiplying and dividing them.

Complex Numbers  Here is a very quick primer on complex numbers and how to manipulate them.


Solving Equations and Inequalities

Solutions and Solution Sets  We introduce some of the basic notation and ideas involved in solving in this section.

Linear Equations  In this section we will solve linear equations, including equations with rational expressions.

Applications of Linear Equations  We will take a quick look at applications of linear equations in this section.

Equations With More Than One Variable  Here we will look at solving equations with more than one variable in them.

Quadratic Equations, Part I  In this section we will start looking at solving quadratic equations.  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 this section.  We will look at completing the square and quadratic formula in this section.

Quadratic Equations : A Summary  We’ll give a procedure for determining which method to use in solving quadratic equations in this section.  We will also take a quick look at the discriminant.

Applications of Quadratic Equations  Here we will revisit some of the applications we saw in the linear application section, only this time they will involve solving a quadratic equation.

Equations Reducible to Quadratic Form  In this section we will solve equations that can be reduced to quadratic in form.

Equations with Radicals  Here we will solve equations with square roots in them.

Linear Inequalities  We will start solving inequalities in this section by looking at linear inequalities.

Polynomial Inequalities  In this section we will look at solving inequalities that contain polynomials.

Rational Inequalities  Here we will solve inequalities involving rational expressions.

Absolute Value Equations  We will officially define absolute value in this section and solve equations that contain absolute value.

Absolute Value Inequalities  We will solve inequalities that involve absolute value in this section.


Graphing and Functions

Graphing  In this section we will introduce the Cartesian coordinate system and most of the basics of graphing equations.

Lines  Here we will review the main ideas from the study of lines including slope and the 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 this section.  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 well as function composition.

Inverse Functions  We will define and find inverse functions in this section.


Common Graphs

Lines, Circles and Piecewise Functions  This section is here only to acknowledge that we’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 functions that 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 also be taking a look at vertical and horizontal asymptotes.


Polynomial Functions

Dividing Polynomials  We’ll review some of the basics of dividing polynomials in this section.

Zeroes/Roots of Polynomials  In this section we’ll define just what zeroes/roots of polynomials 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 rough sketch of some polynomials.

Finding Zeroes of Polynomials  We’ll look at a process that will allow us to find some of 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 fractions and finding the partial fraction decomposition of a rational expression.


Exponential and Logarithm Functions

Exponential Functions  In this section we will introduce exponential functions.  We will be taking a look at some of the properties of exponential functions.

Logarithm Functions  Here we will introduce logarithm functions.  We be looking at how to evaluate logarithms as well as the properties of logarithms.

Solving Exponential Equations  We will be solving equations that contain exponentials 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 exponential functions and an application of logarithms.


Systems of Equations

Linear Systems with Two Variables  In this section we will use systems of two equations and two variables to introduce two of the main methods for solving systems of equations.

Linear Systems with Three Variables  Here we will work a quick example to show how 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 in this 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 special cases to the solutions to systems and how to identify them using the augmented matrix method.

Nonlinear Systems  We will take a quick look at solving nonlinear systems of equations in this section.

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