Paul Dawkins
,Algebra i
Table of Contents
Preface ..................................................................................................................................................................iii
Outline ..................................................................................................................................................................iv
Chapter 1 : Preliminaries ....................................................................................................................................... 1
Section 1-1 : Integer Exponents ................................................................................................................................ 2
Section 1-2 : Rational Exponents .............................................................................................................................. 9
Section 1-3 : Radicals .............................................................................................................................................. 15
Section 1-4 : Polynomials ........................................................................................................................................ 25
Section 1-5 : Factoring Polynomials ........................................................................................................................ 31
Section 1-6 : Rational Expressions .......................................................................................................................... 41
Section 1-7 : Complex Numbers .............................................................................................................................. 52
Chapter 2 : Solving Equations and Inequalities .....................................................................................................59
Section 2-1 : Solutions and Solution Sets ................................................................................................................ 61
Section 2-2 : Linear Equations ................................................................................................................................. 65
Section 2- 3 : Applications of Linear Equations ....................................................................................................... 73
Section 2-4 : Equations With More Than One Variable .......................................................................................... 83
Section 2-5 : Quadratic Equations - Part I ............................................................................................................... 87
Section 2-6 : Quadratic Equations - Part II .............................................................................................................. 95
Section 2-7 : Quadratic Equations : A Summary ................................................................................................... 106
Section 2-8 : Applications of Quadratic Equations ................................................................................................ 109
Section 2-9 : Equations Reducible to Quadratic in Form ...................................................................................... 113
Section 2-10 : Equations with Radicals ................................................................................................................. 118
Section 2-11 : Linear Inequalities .......................................................................................................................... 125
Section 2-12 : Polynomial Inequalities .................................................................................................................. 132
Section 2-13 : Rational Inequalities....................................................................................................................... 138
Section 2-14 : Absolute Value Equations .............................................................................................................. 143
Section 2-15 : Absolute Value Inequalities ........................................................................................................... 150
Chapter 3 : Graphing and Functions ...................................................................................................................155
Section 3-1 : Graphing ........................................................................................................................................... 156
Section 3-2 : Lines ................................................................................................................................................. 163
Section 3-3 : Circles ............................................................................................................................................... 173
Section 3-4 : The Definition of a Function............................................................................................................. 179
Section 3-5 : Graphing Functions .......................................................................................................................... 193
Section 3-6 : Combining Functions........................................................................................................................ 197
Section 3-7 : Inverse Functions ............................................................................................................................. 204
Chapter 4 : Common Graphs ..............................................................................................................................211
Section 4-1 : Lines, Circles and Piecewise Functions............................................................................................. 212
Section 4-1 : Parabolas .......................................................................................................................................... 213
Section 4-3 : Ellipses .............................................................................................................................................. 223
Section 4-4 : Hyperbolas ....................................................................................................................................... 227
Section 4-5 : Miscellaneous Functions .................................................................................................................. 231
Section 4- 6 : Transformations .............................................................................................................................. 234
Section 4-7 : Symmetry ......................................................................................................................................... 240
Section 4-8 : Rational Functions............................................................................................................................ 245
Chapter 5 : Polynomial Functions .......................................................................................................................252
© 2018 Paul Dawkins http://tutorial.math.lamar.edu
,Algebra ii
Section 5-1 : Dividing Polynomials ........................................................................................................................ 253
Section 5-2 : Zeroes/Roots of Polynomials ........................................................................................................... 258
Section 5-3 : Graphing Polynomials ...................................................................................................................... 264
Section 5-4 : Finding Zeroes of Polynomials ......................................................................................................... 272
Section 5-5 : Partial Fractions ............................................................................................................................... 280
Chapter 6 : Exponential and Logarithm Functions ..............................................................................................288
Section 6-1 : Exponential Functions ...................................................................................................................... 289
Section 6-2 : Logarithm Functions......................................................................................................................... 294
Section 6-3 : Solving Exponential Equations ......................................................................................................... 304
Section 6-4 : Solving Logarithm Equations ............................................................................................................ 311
Section 6-5 : Applications ...................................................................................................................................... 317
Chapter 7 : Systems of Equations .......................................................................................................................324
Section 7-1 : Linear Systems with Two Variables .................................................................................................. 325
Section 7-2 : Linear Systems with Three Variables ............................................................................................... 334
Section 7-3 : Augmented Matrices........................................................................................................................ 338
Section 7-4 : More on the Augmented Matrix ...................................................................................................... 347
Section 7-5 : Nonlinear Systems ........................................................................................................................... 353
© 2018 Paul Dawkins http://tutorial.math.lamar.edu
, Algebra iii
Preface
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.
© 2018 Paul Dawkins http://tutorial.math.lamar.edu
,Algebra i
Table of Contents
Preface ..................................................................................................................................................................iii
Outline ..................................................................................................................................................................iv
Chapter 1 : Preliminaries ....................................................................................................................................... 1
Section 1-1 : Integer Exponents ................................................................................................................................ 2
Section 1-2 : Rational Exponents .............................................................................................................................. 9
Section 1-3 : Radicals .............................................................................................................................................. 15
Section 1-4 : Polynomials ........................................................................................................................................ 25
Section 1-5 : Factoring Polynomials ........................................................................................................................ 31
Section 1-6 : Rational Expressions .......................................................................................................................... 41
Section 1-7 : Complex Numbers .............................................................................................................................. 52
Chapter 2 : Solving Equations and Inequalities .....................................................................................................59
Section 2-1 : Solutions and Solution Sets ................................................................................................................ 61
Section 2-2 : Linear Equations ................................................................................................................................. 65
Section 2- 3 : Applications of Linear Equations ....................................................................................................... 73
Section 2-4 : Equations With More Than One Variable .......................................................................................... 83
Section 2-5 : Quadratic Equations - Part I ............................................................................................................... 87
Section 2-6 : Quadratic Equations - Part II .............................................................................................................. 95
Section 2-7 : Quadratic Equations : A Summary ................................................................................................... 106
Section 2-8 : Applications of Quadratic Equations ................................................................................................ 109
Section 2-9 : Equations Reducible to Quadratic in Form ...................................................................................... 113
Section 2-10 : Equations with Radicals ................................................................................................................. 118
Section 2-11 : Linear Inequalities .......................................................................................................................... 125
Section 2-12 : Polynomial Inequalities .................................................................................................................. 132
Section 2-13 : Rational Inequalities....................................................................................................................... 138
Section 2-14 : Absolute Value Equations .............................................................................................................. 143
Section 2-15 : Absolute Value Inequalities ........................................................................................................... 150
Chapter 3 : Graphing and Functions ...................................................................................................................155
Section 3-1 : Graphing ........................................................................................................................................... 156
Section 3-2 : Lines ................................................................................................................................................. 163
Section 3-3 : Circles ............................................................................................................................................... 173
Section 3-4 : The Definition of a Function............................................................................................................. 179
Section 3-5 : Graphing Functions .......................................................................................................................... 193
Section 3-6 : Combining Functions........................................................................................................................ 197
Section 3-7 : Inverse Functions ............................................................................................................................. 204
Chapter 4 : Common Graphs ..............................................................................................................................211
Section 4-1 : Lines, Circles and Piecewise Functions............................................................................................. 212
Section 4-1 : Parabolas .......................................................................................................................................... 213
Section 4-3 : Ellipses .............................................................................................................................................. 223
Section 4-4 : Hyperbolas ....................................................................................................................................... 227
Section 4-5 : Miscellaneous Functions .................................................................................................................. 231
Section 4- 6 : Transformations .............................................................................................................................. 234
Section 4-7 : Symmetry ......................................................................................................................................... 240
Section 4-8 : Rational Functions............................................................................................................................ 245
Chapter 5 : Polynomial Functions .......................................................................................................................252
© 2018 Paul Dawkins http://tutorial.math.lamar.edu
,Algebra ii
Section 5-1 : Dividing Polynomials ........................................................................................................................ 253
Section 5-2 : Zeroes/Roots of Polynomials ........................................................................................................... 258
Section 5-3 : Graphing Polynomials ...................................................................................................................... 264
Section 5-4 : Finding Zeroes of Polynomials ......................................................................................................... 272
Section 5-5 : Partial Fractions ............................................................................................................................... 280
Chapter 6 : Exponential and Logarithm Functions ..............................................................................................288
Section 6-1 : Exponential Functions ...................................................................................................................... 289
Section 6-2 : Logarithm Functions......................................................................................................................... 294
Section 6-3 : Solving Exponential Equations ......................................................................................................... 304
Section 6-4 : Solving Logarithm Equations ............................................................................................................ 311
Section 6-5 : Applications ...................................................................................................................................... 317
Chapter 7 : Systems of Equations .......................................................................................................................324
Section 7-1 : Linear Systems with Two Variables .................................................................................................. 325
Section 7-2 : Linear Systems with Three Variables ............................................................................................... 334
Section 7-3 : Augmented Matrices........................................................................................................................ 338
Section 7-4 : More on the Augmented Matrix ...................................................................................................... 347
Section 7-5 : Nonlinear Systems ........................................................................................................................... 353
© 2018 Paul Dawkins http://tutorial.math.lamar.edu
, Algebra iii
Preface
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.
© 2018 Paul Dawkins http://tutorial.math.lamar.edu
