# 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

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 a

^{2}≥ 0, in other words a

^{2}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 a

^{n}is

defined for any real number a and integer exponent n. Later you will learn that the

laws of exponents still apply for a

^{r}, 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 (ft

^{2}) or square

centimeters (cm

^{2}), and a volume would be measured in cubic feet (ft

^{3}) or cubic

centimeters (cm

^{3}). 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

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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