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Algebra
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 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 when 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 (and brief description) of the material that is in this set of notes.
Preliminaries - In this chapter we will do a quick review of some topics that are absolutely essential to being successful in an Algebra class. We review exponents (integer and rational), radicals, polynomials, factoring polynomials, rational expressions and complex numbers.
Integer Exponents – In this section we will start looking at exponents. We will give the basic properties of exponents and illustrate some of the common mistakes students make in working with exponents. Examples in this section we will be restricted to integer exponents. Rational exponents will be discussed in the next section.
Rational Exponents – In this section we will define what we mean by a rational exponent and extend the properties from the previous section to rational exponents. We will also discuss how to evaluate numbers raised to a rational exponent.
Radicals – In this section we will define radical notation and relate radicals to rational exponents. We will also give the properties of radicals and some of the common mistakes students often make with radicals. We will also define simplified radical form and show how to rationalize the denominator.
Polynomials – In this section we will introduce the basics of polynomials a topic that will appear throughout this course. We will define the degree of a polynomial and discuss how to add, subtract and multiply polynomials.
Factoring Polynomials – In this section we look at factoring polynomials a topic that will appear in pretty much every chapter in this course and so is vital that you understand it. We will discuss factoring out the greatest common factor, factoring by grouping, factoring quadratics and factoring polynomials with degree greater than 2.
Rational Expressions – In this section we will define rational expressions. We will discuss how to reduce a rational expression lowest terms and how to add, subtract, multiply and divide rational expressions.
Complex Numbers – In this section we give a very quick primer on complex numbers including standard form, adding, subtracting, multiplying and dividing them.
Rational Exponents – In this section we will define what we mean by a rational exponent and extend the properties from the previous section to rational exponents. We will also discuss how to evaluate numbers raised to a rational exponent.
Radicals – In this section we will define radical notation and relate radicals to rational exponents. We will also give the properties of radicals and some of the common mistakes students often make with radicals. We will also define simplified radical form and show how to rationalize the denominator.
Polynomials – In this section we will introduce the basics of polynomials a topic that will appear throughout this course. We will define the degree of a polynomial and discuss how to add, subtract and multiply polynomials.
Factoring Polynomials – In this section we look at factoring polynomials a topic that will appear in pretty much every chapter in this course and so is vital that you understand it. We will discuss factoring out the greatest common factor, factoring by grouping, factoring quadratics and factoring polynomials with degree greater than 2.
Rational Expressions – In this section we will define rational expressions. We will discuss how to reduce a rational expression lowest terms and how to add, subtract, multiply and divide rational expressions.
Complex Numbers – In this section we give a very quick primer on complex numbers including standard form, adding, subtracting, multiplying and dividing them.
Solving Equations and Inequalities - In this chapter we will look at one of the most important topics of the class. The ability to solve equations and inequalities is vital to surviving this class and many of the later math classes you might take. We will discuss solving linear and quadratic equations as well as applications. In addition, we will discuss solving polynomial and rational inequalities as well as absolute value equations and inequalities.
Solutions and Solution Sets – In this section we introduce some of the basic notation and ideas involved in solving equations and inequalities. We define solutions for equations and inequalities and solution sets.
Linear Equations – In this section we give a process for solving linear equations, including equations with rational expressions, and we illustrate the process with several examples. In addition, we discuss a subtlety involved in solving equations that students often overlook.
Applications of Linear Equations – In this section we discuss a process for solving applications in general although we will focus only on linear equations here. We will work applications in pricing, distance/rate problems, work rate problems and mixing problems.
Equations With More Than One Variable – In this section we will look at solving equations with more than one variable in them. These equations will have multiple variables in them and we will be asked to solve the equation for one of the variables. This is something that we will be asked to do on a fairly regular basis.
Quadratic Equations, Part I – In this section we will start looking at solving quadratic equations. Specifically, we will look at factoring and the square root property in this section.
Quadratic Equations, Part II – In this section we will continue solving quadratic equations. We will use completing the square to solve quadratic equations in this section and use that to derive the quadratic formula. The quadratic formula is a quick way that will allow us to quickly solve any quadratic equation.
Quadratic Equations : A Summary – In this section we will summarize the topics from the last two sections. We will give a procedure for determining which method to use in solving quadratic equations and we will define the discriminant which will allow us to quickly determine what kind of solutions we will get from solving a quadratic equation.
Applications of Quadratic Equations – In this section we will revisit some of the applications we saw in the linear application section, only this time they will involve solving a quadratic equation. Included are examples in distance/rate problems and work rate problems.
Equations Reducible to Quadratic Form – Not all equations are in what we generally consider quadratic equations. However, some equations, with a proper substitution can be turned into a quadratic equation. These types of equations are called quadratic in form. In this section we will solve this type of equation.
Equations with Radicals – In this section we will discuss how to solve equations with square roots in them. As we will see we will need to be very careful with the potential solutions we get as the process used in solving these equations can lead to values that are not, in fact, solutions to the equation.
Linear Inequalities – In this section we will start solving inequalities. We will concentrate on solving linear inequalities in this section (both single and double inequalities). We will also introduce interval notation.
Polynomial Inequalities – In this section we will continue solving inequalities. However, in this section we move away from linear inequalities and move on to solving inequalities that involve polynomials of degree at least 2.
Rational Inequalities – We continue solving inequalities in this section. We now will solve inequalities that involve rational expressions, although as we’ll see the process here is pretty much identical to the process used when solving inequalities with polynomials.
Absolute Value Equations – In this section we will give a geometric as well as a mathematical definition of absolute value. We will then proceed to solve equations that involve an absolute value. We will also work an example that involved two absolute values.
Absolute Value Inequalities – In this final section of the Solving chapter we will solve inequalities that involve absolute value. As we will see the process for solving inequalities with a \(<\) (i.e. a less than) is very different from solving an inequality with a \(>\) (i.e. greater than).
Linear Equations – In this section we give a process for solving linear equations, including equations with rational expressions, and we illustrate the process with several examples. In addition, we discuss a subtlety involved in solving equations that students often overlook.
Applications of Linear Equations – In this section we discuss a process for solving applications in general although we will focus only on linear equations here. We will work applications in pricing, distance/rate problems, work rate problems and mixing problems.
Equations With More Than One Variable – In this section we will look at solving equations with more than one variable in them. These equations will have multiple variables in them and we will be asked to solve the equation for one of the variables. This is something that we will be asked to do on a fairly regular basis.
Quadratic Equations, Part I – In this section we will start looking at solving quadratic equations. Specifically, we will look at factoring and the square root property in this section.
Quadratic Equations, Part II – In this section we will continue solving quadratic equations. We will use completing the square to solve quadratic equations in this section and use that to derive the quadratic formula. The quadratic formula is a quick way that will allow us to quickly solve any quadratic equation.
Quadratic Equations : A Summary – In this section we will summarize the topics from the last two sections. We will give a procedure for determining which method to use in solving quadratic equations and we will define the discriminant which will allow us to quickly determine what kind of solutions we will get from solving a quadratic equation.
Applications of Quadratic Equations – In this section we will revisit some of the applications we saw in the linear application section, only this time they will involve solving a quadratic equation. Included are examples in distance/rate problems and work rate problems.
Equations Reducible to Quadratic Form – Not all equations are in what we generally consider quadratic equations. However, some equations, with a proper substitution can be turned into a quadratic equation. These types of equations are called quadratic in form. In this section we will solve this type of equation.
Equations with Radicals – In this section we will discuss how to solve equations with square roots in them. As we will see we will need to be very careful with the potential solutions we get as the process used in solving these equations can lead to values that are not, in fact, solutions to the equation.
Linear Inequalities – In this section we will start solving inequalities. We will concentrate on solving linear inequalities in this section (both single and double inequalities). We will also introduce interval notation.
Polynomial Inequalities – In this section we will continue solving inequalities. However, in this section we move away from linear inequalities and move on to solving inequalities that involve polynomials of degree at least 2.
Rational Inequalities – We continue solving inequalities in this section. We now will solve inequalities that involve rational expressions, although as we’ll see the process here is pretty much identical to the process used when solving inequalities with polynomials.
Absolute Value Equations – In this section we will give a geometric as well as a mathematical definition of absolute value. We will then proceed to solve equations that involve an absolute value. We will also work an example that involved two absolute values.
Absolute Value Inequalities – In this final section of the Solving chapter we will solve inequalities that involve absolute value. As we will see the process for solving inequalities with a \(<\) (i.e. a less than) is very different from solving an inequality with a \(>\) (i.e. greater than).
Graphing and Functions - In this chapter we’ll look at two very important topics in an Algebra class. First, we will start discussing graphing equations by introducing the Cartesian (or Rectangular) coordinates system and illustrating use of the coordinate system to graph lines and circles. We will also formally define a function and discuss graph functions and combining functions. We will also discuss inverse functions.
Graphing – In this section we will introduce the Cartesian (or Rectangular) coordinate system. We will define/introduce ordered pairs, coordinates, quadrants, and x and y-intercepts. We will illustrate these concepts with a couple of quick examples
Lines – In this section we will discuss graphing lines. We will introduce the concept of slope and discuss how to find it from two points on the line. In addition, we will introduce the standard form of the line as well as the point-slope form and slope-intercept form of the line. We will finish off the section with a discussion on parallel and perpendicular lines.
Circles – In this section we discuss graphing circles. We introduce the standard form of the circle and show how to use completing the square to put an equation of a circle into standard form.
The Definition of a Function – In this section we will formally define relations and functions. We also give a “working definition” of a function to help understand just what a function is. We introduce function notation and work several examples illustrating how it works. We also define the domain and range of a function. In addition, we introduce piecewise functions in this section.
Graphing Functions – In this section we discuss graphing functions including several examples of graphing piecewise functions.
Combining functions – In this section we will discuss how to add, subtract, multiply and divide functions. In addition, we introduce the concept of function composition.
Inverse Functions – In this section we define one-to-one and inverse functions. We also discuss a process we can use to find an inverse function and verify that the function we get from this process is, in fact, an inverse function.
Lines – In this section we will discuss graphing lines. We will introduce the concept of slope and discuss how to find it from two points on the line. In addition, we will introduce the standard form of the line as well as the point-slope form and slope-intercept form of the line. We will finish off the section with a discussion on parallel and perpendicular lines.
Circles – In this section we discuss graphing circles. We introduce the standard form of the circle and show how to use completing the square to put an equation of a circle into standard form.
The Definition of a Function – In this section we will formally define relations and functions. We also give a “working definition” of a function to help understand just what a function is. We introduce function notation and work several examples illustrating how it works. We also define the domain and range of a function. In addition, we introduce piecewise functions in this section.
Graphing Functions – In this section we discuss graphing functions including several examples of graphing piecewise functions.
Combining functions – In this section we will discuss how to add, subtract, multiply and divide functions. In addition, we introduce the concept of function composition.
Inverse Functions – In this section we define one-to-one and inverse functions. We also discuss a process we can use to find an inverse function and verify that the function we get from this process is, in fact, an inverse function.
Common Graphs - In this chapter we will look at graphing some of the more common functions you might be asked to graph. We graph parabolas, ellipses, hyperbolas and rational functions in this chapter. We will also look at transformations of functions and introduce the concept of symmetry.
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 – In this section we will be graphing parabolas. We introduce the vertex and axis of symmetry for a parabola and give a process for graphing parabolas. We also illustrate how to use completing the square to put the parabola into the form \(f(x)=a(x-h)^{2}+k\).
Ellipses – In this section we will graph ellipses. We introduce the standard form of an ellipse and how to use it to quickly graph an ellipse.
Hyperbolas – In this section we will graph hyperbolas. We introduce the standard form of a hyperbola and how to use it to quickly graph a hyperbola.
Miscellaneous Functions – In this section we will graph a couple of common functions that don’t really take all that much work to do but will be needed in later sections. We’ll be looking at the constant function, square root, absolute value and a simple cubic function.
Transformations – In this section we will be looking at vertical and horizontal shifts of graphs as well as reflections of graphs about the \(x\) and \(y\)-axis. Collectively these are often called transformations and if we understand them they can often be used to allow us to quickly graph some fairly complicated functions.
Symmetry – In this section we introduce the idea of symmetry. We discuss symmetry about the x-axis, y-axis and the origin and we give methods for determining what, if any symmetry, a graph will have without having to actually graph the function.
Rational Functions – In this section we will discuss a process for graphing rational functions. We will also introduce the ideas of vertical and horizontal asymptotes as well as how to determine if the graph of a rational function will have them.
Parabolas – In this section we will be graphing parabolas. We introduce the vertex and axis of symmetry for a parabola and give a process for graphing parabolas. We also illustrate how to use completing the square to put the parabola into the form \(f(x)=a(x-h)^{2}+k\).
Ellipses – In this section we will graph ellipses. We introduce the standard form of an ellipse and how to use it to quickly graph an ellipse.
Hyperbolas – In this section we will graph hyperbolas. We introduce the standard form of a hyperbola and how to use it to quickly graph a hyperbola.
Miscellaneous Functions – In this section we will graph a couple of common functions that don’t really take all that much work to do but will be needed in later sections. We’ll be looking at the constant function, square root, absolute value and a simple cubic function.
Transformations – In this section we will be looking at vertical and horizontal shifts of graphs as well as reflections of graphs about the \(x\) and \(y\)-axis. Collectively these are often called transformations and if we understand them they can often be used to allow us to quickly graph some fairly complicated functions.
Symmetry – In this section we introduce the idea of symmetry. We discuss symmetry about the x-axis, y-axis and the origin and we give methods for determining what, if any symmetry, a graph will have without having to actually graph the function.
Rational Functions – In this section we will discuss a process for graphing rational functions. We will also introduce the ideas of vertical and horizontal asymptotes as well as how to determine if the graph of a rational function will have them.
Polynomial Functions - In this chapter we will take a more detailed look at polynomial functions. We will discuss dividing polynomials, finding zeroes of polynomials and sketching the graph of polynomials. We will also look at partial fractions (even though this doesn’t really involve polynomial functions).
Dividing Polynomials – In this section we’ll review some of the basics of dividing polynomials. We will define the remainder and divisor used in the division process and introduce the idea of synthetic division. We will also give the Division Algorithm.
Zeroes/Roots of Polynomials – In this section we’ll define the zero or root of a polynomial and whether or not it is a simple root or has multiplicity \(k\). We will also give the Fundamental Theorem of Algebra and The Factor Theorem as well as a couple of other useful Facts.
Graphing Polynomials – In this section we will give a process that will allow us to get a rough sketch of the graph of some polynomials. We discuss how to determine the behavior of the graph at \(x\)-intercepts and the leading coefficient test to determine the behavior of the graph as we allow x to increase and decrease without bound.
Finding Zeroes of Polynomials – As we saw in the previous section in order to sketch the graph of a polynomial we need to know what it’s zeroes are. However, if we are not able to factor the polynomial we are unable to do that process. So, in this section we’ll look at a process using the Rational Root Theorem 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. What we will be asking here is what “smaller” rational expressions did we add and/or subtract to get the given rational expression. This is a process that has a lot of uses in some later math classes. It can show up in Calculus and Differential Equations for example.
Zeroes/Roots of Polynomials – In this section we’ll define the zero or root of a polynomial and whether or not it is a simple root or has multiplicity \(k\). We will also give the Fundamental Theorem of Algebra and The Factor Theorem as well as a couple of other useful Facts.
Graphing Polynomials – In this section we will give a process that will allow us to get a rough sketch of the graph of some polynomials. We discuss how to determine the behavior of the graph at \(x\)-intercepts and the leading coefficient test to determine the behavior of the graph as we allow x to increase and decrease without bound.
Finding Zeroes of Polynomials – As we saw in the previous section in order to sketch the graph of a polynomial we need to know what it’s zeroes are. However, if we are not able to factor the polynomial we are unable to do that process. So, in this section we’ll look at a process using the Rational Root Theorem 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. What we will be asking here is what “smaller” rational expressions did we add and/or subtract to get the given rational expression. This is a process that has a lot of uses in some later math classes. It can show up in Calculus and Differential Equations for example.
Exponential and Logarithm Functions - In this chapter we will introduce two very important functions in many areas : the exponential and logarithm functions. We will look at their basic properties, applications and solving equations involving the two functions. If you are in a field that takes you into the sciences or engineering then you will be running into both of these functions.
Exponential Functions – In this section we will introduce exponential functions. We will give some of the basic properties and graphs of exponential functions. We will also discuss what many people consider to be the exponential function, \(f(x) = {\bf e}^{x}\).
Logarithm Functions – In this section we will introduce logarithm functions. We give the basic properties and graphs of logarithm functions. In addition, we discuss how to evaluate some basic logarithms including the use of the change of base formula. We will also discuss the common logarithm, \(\log(x)\), and the natural logarithm, \(\ln(x)\).
Solving Exponential Equations – In this section we will discuss a couple of methods for solving equations that contain exponentials.
Solving Logarithm Equations – In this section we will discuss a couple of methods for solving equations that contain logarithms. Also, as we’ll see, with one of the methods we will need to be careful of the results of the method as it is always possible that the method gives values that are, in fact, not solutions to the equation.
Applications – In this section we will look at a couple of applications of exponential functions and an application of logarithms. We look at compound interest, exponential growth and decay and earthquake intensity.
Logarithm Functions – In this section we will introduce logarithm functions. We give the basic properties and graphs of logarithm functions. In addition, we discuss how to evaluate some basic logarithms including the use of the change of base formula. We will also discuss the common logarithm, \(\log(x)\), and the natural logarithm, \(\ln(x)\).
Solving Exponential Equations – In this section we will discuss a couple of methods for solving equations that contain exponentials.
Solving Logarithm Equations – In this section we will discuss a couple of methods for solving equations that contain logarithms. Also, as we’ll see, with one of the methods we will need to be careful of the results of the method as it is always possible that the method gives values that are, in fact, not solutions to the equation.
Applications – In this section we will look at a couple of applications of exponential functions and an application of logarithms. We look at compound interest, exponential growth and decay and earthquake intensity.
Systems of Equations - In this chapter we will take a look at solving systems of equations. We will solve linear as well as nonlinear systems of equations. We will also take a quick look at using augmented matrices to solve linear systems of equations.
Linear Systems with Two Variables – In this section we will solve systems of two equations and two variables. We will use the method of substitution and method of elimination to solve the systems in this section. We will also introduce the concepts of inconsistent systems of equations and dependent systems of equations.
Linear Systems with Three Variables – In this section we will work a couple of quick examples illustrating how to use the method of substitution and method of elimination introduced in the previous section as they apply to systems of three equations.
Augmented Matrices – In this section we will look at another method for solving systems. We will introduce the concept of an augmented matrix. This will allow us to use the method of Gauss-Jordan elimination to solve systems of equations. We will use the method with systems of two equations and systems of three equations.
More on the Augmented Matrix – In this section we will revisit the cases of inconsistent and dependent solutions to systems and how to identify them using the augmented matrix method.
Nonlinear Systems – In this section we will take a quick look at solving nonlinear systems of equations. A nonlinear system of equations is a system in which at least one of the equations is not linear, i.e. has degree of two or more. Note as well that the discussion here does not cover all the possible solution methods for nonlinear systems. Solving nonlinear systems is often a much more involved process than solving linear systems.
Linear Systems with Three Variables – In this section we will work a couple of quick examples illustrating how to use the method of substitution and method of elimination introduced in the previous section as they apply to systems of three equations.
Augmented Matrices – In this section we will look at another method for solving systems. We will introduce the concept of an augmented matrix. This will allow us to use the method of Gauss-Jordan elimination to solve systems of equations. We will use the method with systems of two equations and systems of three equations.
More on the Augmented Matrix – In this section we will revisit the cases of inconsistent and dependent solutions to systems and how to identify them using the augmented matrix method.
Nonlinear Systems – In this section we will take a quick look at solving nonlinear systems of equations. A nonlinear system of equations is a system in which at least one of the equations is not linear, i.e. has degree of two or more. Note as well that the discussion here does not cover all the possible solution methods for nonlinear systems. Solving nonlinear systems is often a much more involved process than solving linear systems.