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.
- 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.
- 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.
- 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.
- 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.
Preliminaries
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.
Real Exponents This is a short acknowledgment that the
exponent properties from the previous 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 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.