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College Algebra

Contents

Preface iii
   
Summary of Course Components ix
   
1 Real numbers and fundamentals of algebra and geometry 1
   
2 Polynomials, factoring, roots and exponents 4
   
3 Graphing; Linear, rational and quadratic equations 8
   
4 Complex numbers; Solutions to quadratic and radical equations 12
   
5 Inequalities; Graphs of lines; Circles 18
   
6 Graphs; Symmetry and properties of functions 23
   
7 Graphing techniques 28
   
8 Transformations and mathematical models 34
   
9 Quadratic and polynomial functions and their graphs 38
   
10 Polynomial and rational inequalities; Real and complex zeros of polynomials 44
   
11 Composite, one-to-one and inverse functions; Exponential functions 48
   
12 Logarithmic functions; Logarithmic and exponential equations 53
   
13 Exponential growth and decay; Compound interest 58
   
14 Systems of equations: Substitution, elimination, and introduction to matrices 62
   
15 Systems of equations: Determinants and matrices 68
   
16 Systems of equations: Linear programming 74
   
Appendix: First steps with your graphing calculator 77

1.Real numbers and fundamentals of
algebra and geometry



Reading Assignment

Chapter R, Sections R.1, R.2 and R.3, Pages 1–33

Practice Assignments

• Section R.1, Pages 14–15: 9, 25, 31, 35, 41, 45, 51, 55, 61, 65, (Do 61 and 65 without your
calculator), 71

• Section R.2, Pages 26–27: 21, 25, 31, 35, 45, 51, 65, 71, 75, 85, 91, 95, 101, 115, 120

• Section R.3, Page 33: 11, 15, 21, 25, 31, 35, 41, 45

Objectives

In your first lesson, you review basic material covering sets, set notation, real numbers and
some fundamentals of algebra and geometry. The material should not be new to you and
the reading assignment should not be difficult. As you read, you should refresh your
mathematical vocabulary. Note that most of the important words are stressed in the text by
bold faced type. You need not mark your text with a highlighter pen as the author has done
this for you. Pay special attention to the material in the yellow and blue shaded areas.

Your first reading assignment is Chapter R, Sections R.1, R.2, and R.3, pages 1–33 of the
text. You should carefully read and understand all of the many examples, and you should
take a look at all the exercises at the end of each section even if not assigned.

Comments

1. The real number line consists of positive numbers, negative numbers and the number
0, which is neither positive or negative. The term nonnegative numbers refers to the set
of all positive real numbers together with the number 0.

2. A very important use of the absolute value involves the square root. The square root is
defined only for nonnegative numbers. Note that a2 ≥ 0, in other words a2 is
nonnegative, whether a is positive, negative or 0. But denotes the principal root,
which is always nonnegative. In particular, if a is nonnegative, = a. However,
when a is negative, then = −a. A shorthand way of presenting the two
possibilities
is by writing = |a|. In particular, = 3 not ±3. This is shown in
Examples 11 and 12 on page 23 of the text.

3. The laws of exponents are very important to know. Be sure to practice these until you
feel very comfortable using them. In this section of the text, the expression an is
defined for any real number a and integer exponent n. Later you will learn that the
laws of exponents still apply for ar, where r is ANY real number.

4. Be sure to memorize the Pythagorean Theorem. Any time you see a right triangle your
first thought should jump to this theorem. It will come in very handy in mathematical
modeling. The geometry formulas will also be very important in modeling physical
problems. In particular, pay attention to the difference between a measurement of
length (perimeter or circumference), area and volume. For example, a length might be
measured in feet or centimeters, but an area would be in square feet (ft2) or square
centimeters (cm2), and a volume would be measured in cubic feet (ft3) or cubic
centimeters (cm3). You must be careful when you write out your answers on your
homework or tests to include the proper units. For example, if you are asked the
length of a 6 foot stick plus a 4 foot stick, the answer is 10 feet, not 10. Full credit will
be given only if the units are included.

5. To compute the absolute value of a number using your calculator, go to
NUM. Highlight abs( and press . You should see abs( on your calculator
screen. To find |−2|, you would type abs(-2), and the calculator will display 2 after
you press . See the section below on using your calculator.

Graphing calculator

If you are new to using a graphing calculator you should try the exercises in the appendix to
this study guide “First Steps,” with your graphing calculator. There you will find
step-by-step instructions that will get you started. The use of your calculator will be a part
of every assignment and your skill will grow with each lesson. As you progress through the
course you will learn to use the many wonderful capabilities built into the graphing
calculator.

Written Assignment

Your written assignment for Lesson 1 is included at the end of Lesson 2. You will submit the
written assignments for Lessons 1 and 2 together.
 

2
Polynomials, factoring, roots and
exponents



Reading Assignment

Chapter R, Sections R.4, R.5, R.6, R.7 and R.8, Pages 35–82

Practice Assignments

• Section R.4, Page 44: 9, 23, 43, 65, 71, 89

• Section R.5, Pages 52–53: 13, 19, 27, 33, 41, 55, 63, 85, 99

• Section R.6, Pages 56–57: 5, 7, 9, 23

• Section R.7, Pages 66–67: 7, 11, 25, 33, 49, 71

• Section R.8, Pages 74–75: 13, 17, 31, 47, 65, 69

Objectives

In this lesson you review fundamental algebraic operations involving factoring, synthetic
division, roots and exponents. You then use these basic operations to simplify rational
expressions and complex fractions. As usual, you should study all the examples in the text.
You should encounter no new material in this assignment. Look over the remainder of
Chapter R and, for now, just note the scope of the material.

Comments

1. Before starting the detailed reading and study let’s play a little numbers game with
your calculator that is similar to the old card game that starts with “Take a card, any
card.”

(a) Turn on your calculator and set the mode to float to display 10 digits. To do this
press ; make sure Float is highlighted. If it is not, use the arrow keys to
highlight it and press . Then press QUIT to return to the home
screen.

(b) Perform any calculation that gives a positive answer between 1 and 100,000. For
example, you could enter 2+3 then press .

(c) Press .

(d) Press the following keys in this order:

(e) Repeat step (d) 24 times .

(f) Read the answer: It should say 1.618033989

(g) Question: Is it magic?

In performing the operations above, you have converted your calculator into a
computer. The calculator served as the central processor and your brain created the
program, which was the series of keystrokes that you performed. With the aid of your
calculator you have just solved problem 93 on page 68 by showing that no matter what
positive value of x is selected, the continued fraction will eventually converge to a
number whose first ten digits are 1.618033989.

2. You should memorize the information in the yellow shaded boxes on page 39.
Knowing how to multiply squares of binomials “in your head” and recognizing when
you have the difference of two squares will make your life much easier. You will often
encounter these two types of quadratic expressions. Note that this information is
repeated in the factoring section on page 46 in the reverse order.

3. Pay special attention to the domain of rational expressions when you are reducing
them to lowest terms. For example, can be reduced to x − 2. The
expression x − 2 is defined for all real numbers but the original expression is
defined for all real numbers except x = 2. This means they are NOT the same
expression, as they are not defined for the same domain. This is a very important
distinction between the two.

4. You will probably notice that steps needed to perform operations such as addition ,
multiplication and division with rational expressions look very much like those used
to perform the same operations with numerical fractions . You are correct. They do
require the same steps, such as finding a common denominator. If you find yourself
unsure of what to do with rational expressions, just try the same operation with
fractions, carefully noting the steps you take. Then follow the same steps with the
rational expressions.

5. In Example 12 on page 65 you will see an actual mathematical model for computing
total resistance in an electric circuit that contains two resistors connected in parallel.
You will notice that in this problem the variables are quite different from your usual x
and y. The letter R is commonly used to denote resistance. Because there are three
values for resistance, one for each resistor and one for the total resistance in the model,
there are three different R’s in the equation. These values are differentiated by the
subscripts, as in R1 and R2, and the total resistance is denoted by R with no subscript.

6. It has long been customary to rationalize denominators when radicals appear in
denominators of quotients. You should learn this technique as there will be occasions
on which this might come in handy. However, this is no longer necessary most of the
time. You will not be expected to rationalize the denominator in your answers in this
course unless the instructor specifically asks you to do so. Be sure to read the history
of this technique below.

7. In the years BC (before computers) the term “computer” was a job title for a person. A
computer was provided with many volumes of numerical tables and a calculating
machine, usually a mechanical device that could add. The tables included squares of
numbers, cubes, square roots, reciprocals, powers, logarithms, antilogarithms,
trigonometric values and more. In those days the process of computing to 8
decimal places without rationalizing would require dividing 1.414213562 into
1.000000000 where the value for is obtained from a table. It should be obvious that
when is rationalized to , it is considerably easier to calculate 2 into 1.414213562.
With the invention of the electronic computer and calculators, we no longer have to
worry about this problem. Try entering and into your calculator. You will see
that both decimal approximations are the same and take the same very small amount
of time to calculate.

8. Be sure you are comfortable with expressions containing rational exponents and
converting them from exponential form to radical form. You should also pay special
attention to Example 10 on page 73.

Written Assignment 1

Section Pages Problems
R.1 14–15 32
R.2 26–27 68, 92
R.3 33 24, 32, 34
R.4 44 42, 52, 66
R.5 52–53 24, 50, 92
R.6 56–57 6
R.7 66–67 10, 26, 50, 72
R.8 74–75 32, 62, 70

Please review the Course Mechanics section of this study guide. Reminder: Be sure to
include an Independent Study Cover Sheet labeled “Written Assignment 1.”

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