CHAPTER RESOURCES : AUTHOR TIPS Chapter P Prerequisites [ TOP ] As you begin studying algebra, remember that algebra takes time and practice. Although this chapter starts with the basics, most students will be familiar with the material presented here. In this chapter, we review the basic concepts of real numbers, exponents, polynomials, and rational expressions. These concepts were not invented by me, and they were not invented overnight. The real-number system itself was developed over hundreds of years. The definition of rational exponents is another big achievement. Notice how the definition allows you to do arithmetic with exponents. This did not occur by accident. Your instructor may assume that you have some knowledge of basic concepts of algebra and skip some or all of this chapter. In that case, you should take some time to look over the definitions and rules and at least work the problems presented as examples in the text. The beginning of the semester is a good time to form a study group. Introduce yourself to some of your fellow classmates and find three or four who would like to get together on a regular basis and work problems. Students who work with study groups generally do much better in mathematics classes. You might even get your instructor to meet with your study group from time to time. Chapter 1 Equations and Inequalities [ TOP ] In this chapter, we study linear and quadratic equations and inequalities. Since this material is usually covered in high-school algebra, your instructor may skip some of this chapter. In this chapter, you will see the complex-number system, which is an extension of the real-number system. Adding the imaginary numbers to the real-number system makes the system complete in that the solutions to all polynomial equations are found in the complex-number system. At the end of Chapter 1, you will find a set of exercises called Tying It All Together (TIAT). These exercises combine concepts from the present chapter with those of previous chapters. In TIAT exercises for Chapters P-1, we look at the difference between solving equations and simplifying expressions. These ideas are sometimes confused because the exercises for them look so much alike. When simplifying an expression, you are finding another expression that is equivalent to the given expression. When solving equations, you are looking for numbers that satisfy the equations. If you haven't yet formed a study group, now would be a good time to do this. Don't wait until you are doing badly to get started. Be an active algebra student, not a passive one. It is far better to be ahead of the game in the beginning than trying to play catch up at the end. Doing homework problems is the key to success. No instructor can install knowledge into your head in the short time that you are in the classroom. Many schools have free tutoring programs. Find out what is available at your school and take advantage of every opportunity. Chapter 2 Functions and Graphs [ TOP ] I am sure that you have seen graphs showing the rise and fall of the stock market, the growing world population, or the decrease in federal spending (joke). This simple idea of showing the relationship between two variables with a graph is due to Rene Descartes and is the subject of this chapter. The idea of one variable being a function of another is introduced in this chapter. The concept of functions is central to all of algebra, trigonometry, calculus, and most higher-level areas of mathematics. Although the idea of graphing is simple, it can be very tedious to plot lots of points with an acceptable degree of accuracy. With the invention of the graphing calculator (or computer graphing software), the drudgery is taken out of graphing. If you don't already have a graphing calculator, you should get one. Descartes would have given anything for a graphing calculator, for they are to mathematics what word processors are to writing. You wouldn't think of typing a theme paper on a typewriter in this day and age. Of course, a graphing calculator does not do your homework for you. It cannot think, and it can give incorrect or misleading answers. To use a graphing calculator effectively, you still must know what you are doing. Chapter 3 Polynomial and Rational Functions [ TOP ] Section 3.1 is about linear functions, which are first degree polynomial functions. Section 3.2 is about quadratic functions, which are second-degree polynomial functions. For the rest of the chapter, we study the general theory of polynomial functions and equations. In the last section, we look at the rational functions, which are the ratios of the polynomial functions. Look for the relationships between degree of a polynomial function, the shape of its graph, the x-intercepts, and the factors of the polynomial. The fundamental theorem of algebra is discussed in this chapter. With a title like that it must be important. If you are working on Chapter 3, then you should have already developed some good study habits for this course. Taking good notes in class, working on your homework as soon as possible after class, and reading the examples and trying a few problems before the material is discussed in class are great strategies for success. Working examples prior to class also has the advantage of pinpointing any problem areas before your instructor presents the material. Chapter 4 Exponential and Logarithmic Functions [ TOP ] You have probably heard people say that the population or the federal budget is growing exponentially. In this chapter we will study exponential growth and decay and will see some great applications of exponential functions such as carbon dating, compound interest, and Newton's law of cooling. Logarithms were invented to solve exponential equations. They are used to find the rate or the time in exponential growth or decay. Logarithms are also used in describing the pH of a substance and the intensity of an earthquake. The TIAT exercises at the end of this section provide a good review of the types of equations that you should be able to solve at this point. Don't wait until the night before the final exam to start reviewing. Start now by solving these equations. Many graphing calculators have built in equation solvers. They are fun to use, but they are not a substitute for learning how to solve equations by hand. When you solve an equation by hand, you are practicing and reinforcing the basic principles of algebra. After the TIAT exercises, you will find a page showing the graphs of the basic functions of algebra. You should review this page and be sure that you understand why each function has the type of graph that you see. Chapter 5 Systems of Equations and Inequalities [ TOP ] You have seen linear equations in Chapter 3. Here, we look at systems of linear equations. You will learn a variety of methods for solving systems of equations in this chapter and the next. It is good to learn these methods and use them to solve systems of two or three equations by hand. The methods apply to systems of any number of equations, but it can get really tedious solving a system of ten equations with ten unknowns by hand, so the methods of matrices and determinants have been included in many graphing calculators. Because a graphing calculator is limited in memory, you could use a computer program to solve a very large system. Chapter 6 Matrices and Determinants [ TOP ] This chapter is a continuation of the previous chapter. However, in this chapter we concentrate on the methods of solving linear systems that are readily available on graphing calculators and computers. That does not mean that you must use a graphing calculator or computer to study this chapter. In this chapter, we learn how these methods work, not how to push the buttons on a calculator, because the technology will surely change. If you pick up a graphing calculator in the year 2010, you will probably have to learn how to operate it. If you know the mathematics behind what you want to do with technology, you will not have a difficult time adapting to changes in technology. Chapter 7 The Conic Sections [ TOP ] This is probably my favorite chapter. The curves studied in this section are certainly among the most useful and practical applications of mathematics. The orbits of satellites are elliptical, lights use parabolic or elliptical reflectors, and hyperbolic mirrors are used in lenses. Of course, circular objects are everywhere--I just saw advertised on TV a new barbecue grill with a unique parabolic heat reflector. Chapter 8 Sequences, Series, and Probability [ TOP ] This chapter contains some miscellaneous topics that you may or may not get time to study in your course. However, you never know what you will need in the future and there are some very practical applications in this chapter. Finding the monthly payment to pay off a loan is an application of geometric series. The ideas of probability are important in understanding risk as well as being the basis for the study of statistics. So keep this text as a reference book in your personal library--I hope that you have enjoyed studying from it.

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