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College Algebra 2e
SENIOR CONTRIBUTING AUTHOR
JAY ABRAMSON, ARIZONA STATE UNIVERSITY

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Contents
Preface 1
1 Prerequisites 7
Introduction to Prerequisites 7 1.1 Real Numbers: Algebra Essentials 7 1.2 Exponents and Scientific Notation 24 1.3 Radicals and Rational Exponents 39 1.4 Polynomials 50 1.5 Factoring Polynomials 59 1.6 Rational Expressions 68 Chapter Review 76 Exercises 79
2 Equations and Inequalities 83
Introduction to Equations and Inequalities 83 2.1 The Rectangular Coordinate Systems and Graphs 83 2.2 Linear Equations in One Variable 98 2.3 Models and Applications 115 2.4 Complex Numbers 125 2.5 Quadratic Equations 134 2.6 Other Types of Equations 149 2.7 Linear Inequalities and Absolute Value Inequalities 161 Chapter Review 172 Exercises 175
3 Functions 181
Introduction to Functions 181 3.1 Functions and Function Notation 181 3.2 Domain and Range 205 3.3 Rates of Change and Behavior of Graphs 223 3.4 Composition of Functions 239 3.5 Transformation of Functions 255 3.6 Absolute Value Functions 287 3.7 Inverse Functions 295 Chapter Review 310 Exercises 314
4 Linear Functions 323
Introduction to Linear Functions 323 4.1 Linear Functions 323 4.2 Modeling with Linear Functions 360 4.3 Fitting Linear Models to Data 374 Chapter Review 388 Exercises 389
5 Polynomial and Rational Functions 399
Introduction to Polynomial and Rational Functions 399

5.1 Quadratic Functions 400 5.2 Power Functions and Polynomial Functions 419 5.3 Graphs of Polynomial Functions 438 5.4 Dividing Polynomials 460 5.5 Zeros of Polynomial Functions 471 5.6 Rational Functions 484 5.7 Inverses and Radical Functions 508 5.8 Modeling Using Variation 521 Chapter Review 531 Exercises 535
6 Exponential and Logarithmic Functions 541
Introduction to Exponential and Logarithmic Functions 541 6.1 Exponential Functions 542 6.2 Graphs of Exponential Functions 561 6.3 Logarithmic Functions 575 6.4 Graphs of Logarithmic Functions 584 6.5 Logarithmic Properties 606 6.6 Exponential and Logarithmic Equations 618 6.7 Exponential and Logarithmic Models 631 6.8 Fitting Exponential Models to Data 650 Chapter Review 666 Exercises 671
7 Systems of Equations and Inequalities 681
Introduction to Systems of Equations and Inequalities 681 7.1 Systems of Linear Equations: Two Variables 682 7.2 Systems of Linear Equations: Three Variables 701 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables 713 7.4 Partial Fractions 725 7.5 Matrices and Matrix Operations 735 7.6 Solving Systems with Gaussian Elimination 748 7.7 Solving Systems with Inverses 762 7.8 Solving Systems with Cramer's Rule 777 Chapter Review 790 Exercises 793
8 Analytic Geometry 801
Introduction to Analytic Geometry 801 8.1 The Ellipse 802 8.2 The Hyperbola 818 8.3 The Parabola 835 8.4 Rotation of Axes 851 8.5 Conic Sections in Polar Coordinates 866 Chapter Review 877 Exercises 879
9 Sequences, Probability, and Counting Theory 885
Introduction to Sequences, Probability and Counting Theory 885
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9.1 Sequences and Their Notations 885 9.2 Arithmetic Sequences 900 9.3 Geometric Sequences 912 9.4 Series and Their Notations 921 9.5 Counting Principles 935 9.6 Binomial Theorem 946 9.7 Probability 953 Chapter Review 964 Exercises 967
Answer Key 975
Index 1099

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Preface 1
Preface
About OpenStax
OpenStax is part of Rice University, which is a 501(c)(3) nonprofit charitable corporation. As an educational initiative, it's our mission to transform learning so that education works for every student. Through our partnerships with philanthropic organizations and our alliance with other educational resource companies, we're breaking down the most common barriers to learning. Because we believe that everyone should and can have access to knowledge.
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Format You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print.
About College Algebra 2e
College Algebra 2e provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra course. The modular approach and richness of content ensure that the book meets the needs of a variety of courses. College Algebra offers a wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking students to apply what they’ve learned.
Coverage and Scope
In determining the concepts, skills, and topics to cover, we engaged dozens of highly experienced instructors with a range of student audiences. The resulting scope and sequence proceeds logically while allowing for a significant amount of flexibility in instruction.
Chapters 1 and 2 provide both a review and foundation for study of functions that begins in Chapter 3. The authors recognize that while some institutions may find this material a prerequisite, other institutions have told us that they have a cohort that need the prerequisite skills built into the course.
• Chapter 1: Prerequisites • Chapter 2: Equations and Inequalities
Chapters 3-6: The Algebraic Functions
• Chapter 3: Functions • Chapter 4: Linear Functions

2 Preface
• Chapter 5: Polynomial and Rational Functions • Chapter 6: Exponential and Logarithm Functions
Chapters 7-9: Further Study in College Algebra
• Chapter 7: Systems of Equations and Inequalities • Chapter 8: Analytic Geometry • Chapter 9: Sequences, Probability, and Counting Theory
Development Overview
College Algebra 2e is the product of a collaborative effort by a group of dedicated authors, editors, and instructors whose collective passion for this project has resulted in a text that is remarkably unified in purpose and voice. Special thanks is due to our Lead Author, Jay Abramson of Arizona State University, who provided the overall vision for the book and oversaw the development of each and every chapter, drawing up the initial blueprint, reading numerous drafts, and assimilating field reviews into actionable revision plans for our authors and editors.
The collective experience of our author team allowed us to pinpoint the subtopics, exceptions, and individual connections that give students the most trouble. The textbook is therefore replete with well-designed features and highlights, which help students overcome these barriers. As the students read and practice, they are coached in methods of thinking through problems and internalizing mathematical processes.
Accuracy of the Content
We understand that precision and accuracy are imperatives in mathematics, and undertook an accuracy program led by experienced faculty. Examples, art, problems, and solutions were reviewed by dedicated faculty, with a separate team evaluating the answer key and solutions.
The text also benefits from years of usage by thousands of faculty and students. A core aspect of the second edition revision process included consolidating and ensuring consistency with regard to any errata and corrections that have been implemented during the series' extensive usage and incorporation into homework systems.
Changes to the Second Edition
The College Algebra 2e revision focused on mathematical clarity and accuracy as well as inclusivity. Examples, Exercises, and Solutions were reviewed by multiple faculty experts. All improvement suggestions and errata updates, driven by faculty and students from several thousand colleges, were considered and unified across the different formats of the text.
OpenStax and our authors are aware of the difficulties posed by shifting problem and exercise numbers when textbooks are revised. In an effort to make the transition to the 2nd edition as seamless as possible, we have minimized any shifting of exercise numbers.
The revision also focused on supporting inclusive and welcoming learning experiences. The introductory narratives, example and problem contexts, and even many of the names used for fictional people in the text were all reviewed using a diversity, equity, and inclusion framework. Several hundred resulting revisions improve the balance and relevance to the students using the text, while maintaining a variety of applications to diverse careers and academic fields. In particular, explanations of scientific and historical aspects of mathematics have been expanded to include more contributors. For example, the authors added additional historical and multicultural context regarding what is widely known as Pascal’s Triangle, and similarly added details regarding the international process of decoding the Enigma machine (including the role of Polish college students). Several chapter-opening narratives and in-chapter references are completely new, and contexts across all chapters were specifically reviewed for equity in gender representation and connotation.
Finally, prior to the release of this edition, OpenStax published a version to support Corequisite instruction, which is described in more detail below.
Pedagogical Foundations and Features
Learning Objectives
Each chapter is divided into multiple sections (or modules), each of which is organized around a set of learning objectives. The learning objectives are listed explicitly at the beginning of each section and are the focal point of every instructional element
Narrative text
Narrative text is used to introduce key concepts, terms, and definitions, to provide real-world context, and to provide
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Preface 3
transitions between topics and examples. Throughout this book, we rely on a few basic conventions to highlight the most important ideas:
• Key terms are boldfaced, typically when first introduced and/or when formally defined. • Key concepts and definitions are called out in a blue box for easy reference.
Examples
Each learning objective is supported by one or more worked examples that demonstrate the problem-solving approaches that students must master. The multiple Examples model different approaches to the same type of problem or introduce similar problems of increasing complexity. All Examples follow a simple two- or three-part format. The question clearly lays out a mathematical problem to solve. The Solution walks through the steps, usually providing context for the approach — in other words, why the instructor is solving the problem in a specific manner. Finally, the Analysis (for select examples) reflects on the broader implications of the Solution just shown. Examples are followed by a “Try It” question, as explained below.
Figures
College Algebra 2e contains many figures and illustrations, the vast majority of which are graphs and diagrams. Art throughout the text adheres to a clear, understated style, drawing the eye to the most important information in each figure while minimizing visual distractions. Color contrast is employed with discretion to distinguish between the different functions or features of a graph.
Supporting Features
Several elements, each marked by a distinctive icon, contribute to and check understanding. • A How To is a list of steps necessary to solve a certain type of problem. A How To typically precedes an Example that proceeds to demonstrate the steps in action. • A Try It exercise immediately follows an Example or a set of related Examples, providing the student with an immediate opportunity to solve a similar problem. In the PDF and the Web View version of the text, answers to the Try It exercises are located in the Answer Key. • A Q&A may appear at any point in the narrative, but most often follows an Example. This feature pre-empts misconceptions by posing a commonly asked yes/no question, followed by a detailed answer and explanation. • The Media icon appears at the conclusion of each section, just prior to the Section Exercises. This icon marks a list of links to online video tutorials that reinforce the concepts and skills introduced in the section.
While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor were they specifically produced or tailored to accompany College Algebra 2e.
Section Exercises
Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used selectively for guided practice. With over 4600 exercises across the 9 chapters, instructors should have plenty from which to choose. Section Exercises are organized by question type, and generally appear in the following order:
• Verbal questions assess conceptual understanding of key terms and concepts. • Algebraic problems require students to apply algebraic manipulations demonstrated in the section. • Graphical problems assess students’ ability to interpret or produce a graph. • Numeric problems require the student to perform calculations or computations. • Technology problems encourage exploration through use of a graphing utility, either to visualize or verify algebraic
results or to solve problems via an alternative to the methods demonstrated in the section.

4 Preface
• Extensions pose problems more challenging than the Examples demonstrated in the section. They require students to synthesize multiple learning objectives or apply critical thinking to solve complex problems.
• Real-World Applications present realistic problem scenarios from fields such as physics, geology, biology, finance, and the social sciences.
Chapter Review Features
Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that students can use to prepare for exams.
• Key Terms provides a formal definition for each bold-faced term in the chapter. • Key Equations presents a compilation of formulas, theorems, and standard-form equations. • Key Concepts summarizes the most important ideas introduced in each section, linking back to the relevant
Example(s) in case students need to review. • Chapter Review Exercises include 40-80 practice problems that recall the most important concepts from each
section. • Practice Test includes 25-50 problems assessing the most important learning objectives from the chapter. Note
that the practice test is not organized by section, and may be more heavily weighted toward cumulative objectives as opposed to the foundational objectives covered in the opening sections. • Answer Key includes the answers to all Try It exercises and every other exercise from the Section Exercises, Chapter Review Exercises, and Practice Test.
Corequisite Support
Each College Algebra 2e section is paired with a thoughtfully developed, topically aligned skills module that prepares students for the course material. Sharon North (St. Louis Community College) developed a coordinated set of support resources, which provide review, instruction, and practice for algebra students. The author team identified foundational skills and concepts, then mapped them to each module. The corequisite sections include conceptual overviews, worked examples, and guided practice; they incorporate relevant material from OpenStax’s Developmental Math series. They are available as separate, openly accessible downloads from the student and instructor resources pages accompanying the text. And they are also provided in an integrated manner in an alternate version of the text, College Algebra 2e with Corequisite Support.
Community Hubs
OpenStax partners with the Institute for the Study of Knowledge Management in Education (ISKME) to offer Community Hubs on OER Commons—a platform for instructors to share community-created resources that support OpenStax books, free of charge. Through our Community Hubs, instructors can upload their own materials or download resources to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant course content. We encourage instructors to join the hubs for the subjects most relevant to your teaching and research as an opportunity both to enrich your courses and to engage with other faculty. To reach the Community Hubs, visit www.oercommons.org/hubs/openstax.
Technology partners
As allies in making high-quality learning materials accessible, our technology partners offer optional low-cost tools that are integrated with OpenStax books. To access the technology options for your text, visit your book page on openstax.org.
Additional Resources
Student and Instructor Resources We’ve compiled additional resources for both students and instructors, including Getting Started Guides, an instructor solution manual, and PowerPoint slides. Instructor resources require a verified instructor account, which can be requested on your openstax.org log-in. Take advantage of these resources to supplement your OpenStax book.
About the Authors
Senior Contributing Author
Jay Abramson, Arizona State University Jay Abramson has been teaching Precalculus for over 35 years, the last 20 at Arizona State University, where he is a principal lecturer in the School of Mathematics and Statistics. His accomplishments at ASU include co-developing the university’s first hybrid and online math courses as well as an extensive library of video lectures and tutorials. In addition, he has served as a contributing author for two of Pearson Education’s math programs, NovaNet Precalculus and Trigonometry. Prior to coming to ASU, Jay taught at Texas State Technical College and Amarillo College. He received Teacher of the Year awards at both institutions.
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Preface 5
Contributing Authors
Valeree Falduto, Palm Beach State College Rachael Gross, Towson University David Lippman, Pierce College Melonie Rasmussen, Pierce College Rick Norwood, East Tennessee State University Nicholas Belloit, Florida State College Jacksonville Jean-Marie Magnier, Springfield Technical Community College Harold Whipple Christina Fernandez
Reviewers
Phil Clark, Scottsdale Community College Michael Cohen, Hofstra University Matthew Goodell, SUNY Ulster Lance Hemlow, Raritan Valley Community College Dongrin Kim, Arizona State University Cynthia Landrigan, Erie Community College Wendy Lightheart, Lane Community College Carl Penziul, Tompkins-Cortland Community College Sandra Nite, Texas A&M University Eugenia Peterson, Richard J. Daley College Rhonda Porter, Albany State University Michael Price, University of Oregon William Radulovich, Florida State College Jacksonville Camelia Salajean, City Colleges of Chicago Katy Shields, Oakland Community College Nathan Schrenk, ECPI University Pablo Suarez, Delaware State University Allen Wolmer, Atlanta Jewish Academy
The following faculty contributed to the development of OpenStax Precalculus, the text from which this product was updated and derived. Precalculus Reviewers Nina Alketa, Cecil College Kiran Bhutani, Catholic University of America Brandie Biddy, Cecil College Lisa Blank, Lyme Central School Bryan Blount, Kentucky Wesleyan College Jessica Bolz, The Bryn Mawr School Sheri Boyd, Rollins College Sarah Brewer, Alabama School of Math and Science Charles Buckley, St. Gregory's University Kenneth Crane, Texarkana College Rachel Cywinski, Alamo Colleges Nathan Czuba Srabasti Dutta, Ashford University Kristy Erickson, Cecil College Nicole Fernandez, Georgetown University / Kent State University David French, Tidewater Community College Douglas Furman, SUNY Ulster Erinn Izzo, Nicaragua Christian Academy John Jaffe Jerry Jared, Blue Ridge School Stan Kopec, Mount Wachusett Community College Kathy Kovacs Sara Lenhart, Christopher Newport University Joanne Manville, Bunker Hill Community College Karla McCavit, Albion College

6 Preface
Cynthia McGinnis, Northwest Florida State College Lana Neal, University of Texas at Austin Steven Purtee, Valencia College Alice Ramos, Bethel College Nick Reynolds, Montgomery Community College Amanda Ross, A. A. Ross Consulting and Research, LLC Erica Rutter, Arizona State University Sutandra Sarkar, Georgia State University Willy Schild, Wentworth Institute of Technology Todd Stephen, Cleveland State University Scott Sykes, University of West Georgia Linda Tansil, Southeast Missouri State University John Thomas, College of Lake County Diane Valade, Piedmont Virginia Community College
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1 • Introduction 7

1

PREREQUISITES

Credit: Andreas Kambanls
Chapter Outline
1.1 Real Numbers: Algebra Essentials 1.2 Exponents and Scientific Notation 1.3 Radicals and Rational Exponents 1.4 Polynomials 1.5 Factoring Polynomials 1.6 Rational Expressions
Introduction to Prerequisites
It’s a cold day in Antarctica. In fact, it’s always a cold day in Antarctica. Earth’s southernmost continent, Antarctica experiences the coldest, driest, and windiest conditions known. The coldest temperature ever recorded, over one hundred degrees below zero on the Celsius scale, was recorded by remote satellite. It is no surprise then, that no native human population can survive the harsh conditions. Only explorers and scientists brave the environment for any length of time.
Measuring and recording the characteristics of weather conditions in Antarctica requires a use of different kinds of numbers. For tens of thousands of years, humans have undertaken methods to tally, track, and record numerical information. While we don't know much about their usage, the Lebombo Bone (dated to about 35,000 BCE) and the Ishango Bone (dated to about 20,000 BCE) are among the earliest mathematical artifacts. Found in Africa, their clearly deliberate groupings of notches may have been used to track time, moon cycles, or other information. Performing calculations with them and using the results to make predictions requires an understanding of relationships among numbers. In this chapter, we will review sets of numbers and properties of operations used to manipulate numbers. This understanding will serve as prerequisite knowledge throughout our study of algebra and trigonometry.
1.1 Real Numbers: Algebra Essentials
Learning Objectives
In this section, you will: Classify a real number as a natural, whole, integer, rational, or irrational number. Perform calculations using order of operations. Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity. Evaluate algebraic expressions. Simplify algebraic expressions.

8 1 • Prerequisites

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe

them in set notation as

where the ellipsis (…) indicates that the numbers continue to infinity. The natural

numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the

coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is

the set of natural numbers plus zero:

The set of integers adds the opposites of the natural numbers to the set of whole numbers: It is useful to note that the set of integers is made up of three distinct subsets: negative
integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

The set of rational numbers is written as

Notice from the definition that rational

numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

ⓐ a terminating decimal:

ⓑ or

a repeating decimal:

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

EXAMPLE 1

Writing Integers as Rational Numbers Write each of the following as a rational number.

ⓐ 7 ⓑ 0 ⓒ –8 Solution Write a fraction with the integer in the numerator and 1 in the denominator.

ⓐ

ⓑ

ⓒ

TRY IT #1

Write each of the following as a rational number.
ⓐ 11 ⓑ 3 ⓒ –4

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1.1 • Real Numbers: Algebra Essentials 9

EXAMPLE 2

Identifying Rational Numbers Write each of the following rational numbers as either a terminating or repeating decimal.

ⓐ

ⓑⓒ

Solution

Write each fraction as a decimal by dividing the numerator by the denominator.

ⓐ

ⓑ a repeating decimal

(or 3.0), a terminating decimal

ⓒ

a terminating decimal

TRY IT #2

Write each of the following rational numbers as either a terminating or repeating decimal.
ⓐⓑⓒ

Irrational Numbers
At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even but was something else. Or
a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

EXAMPLE 3

Differentiating Rational and Irrational Numbers Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

ⓐ

ⓑⓒ

ⓓ

Solution

ⓐ

This can be simplified as

ⓑ

Because it is a fraction of integers,

ⓔ

Therefore,

is rational.

is a rational number. Next, simplify and divide.

So, is rational and a repeating decimal.

ⓒ

This cannot be simplified any further. Therefore,

is an irrational number.

ⓓ

Because it is a fraction of integers, is a rational number. Simplify and divide.

So, is rational and a terminating decimal.

ⓔ

is not a terminating decimal. Also note that there is no repeating pattern because the group

of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational

number. It is an irrational number.

10 1 • Prerequisites

TRY IT #3

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

ⓐⓑ

ⓒ

ⓓⓔ

Real Numbers
Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.
The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a oneto-one correspondence. We refer to this as the real number line as shown in Figure 1.

Figure 1 The real number line

EXAMPLE 4

Classifying Real Numbers Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

ⓐ

ⓑ

ⓒ

Solution

ⓓ

ⓔ

ⓐ

is negative and rational. It lies to the left of 0 on the number line.

ⓑ

is positive and irrational. It lies to the right of 0.

ⓒ

is negative and rational. It lies to the left of 0.

ⓓ

is negative and irrational. It lies to the left of 0.

ⓔ

is a repeating decimal so it is rational and positive. It lies to the right of 0.

TRY IT #4

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

ⓐ

ⓑ

ⓒⓓ

ⓔ

Sets of Numbers as Subsets
Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2.

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1.1 • Real Numbers: Algebra Essentials 11

Figure 2 Sets of numbers N: the set of natural numbers W: the set of whole numbers I: the set of integers Q: the set of rational numbers Q´: the set of irrational numbers
Sets of Numbers
The set of natural numbers includes the numbers used for counting: The set of whole numbers is the set of natural numbers plus zero: The set of integers adds the negative natural numbers to the set of whole numbers: The set of rational numbers includes fractions written as
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating:

EXAMPLE 5

Differentiating the Sets of Numbers Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

ⓐ

ⓑⓒ

Solution

ⓓⓔ

N W I Q Q′

a.

X XXX

b.

X

c. d. –6
e. 3.2121121112...

X XX
X

12 1 • Prerequisites

TRY IT #5

Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

ⓐ

ⓑⓒ

ⓓ

ⓔ

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example,

We can raise

any number to any power. In general, the exponential notation means that the number or variable is used as a

factor times.

In this notation, is read as the nth power of or to the where is called the base and is called the exponent. A

term in exponential notation may be part of a mathematical expression, which is a combination of numbers and

operations. For example,

is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.
Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.
The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.
Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify as 16.

Next, perform multiplication or division, left to right. Lastly, perform addition or subtraction, left to right.

Therefore, For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.
Order of Operations Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the
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1.1 • Real Numbers: Algebra Essentials 13

acronym PEMDAS: P(arentheses) E(xponents) M(ultiplication) and D(ivision) A(ddition) and S(ubtraction)
...
HOW TO
Given a mathematical expression, simplify it using the order of operations. Step 1. Simplify any expressions within grouping symbols. Step 2. Simplify any expressions containing exponents or radicals. Step 3. Perform any multiplication and division in order, from left to right. Step 4. Perform any addition and subtraction in order, from left to right.

EXAMPLE 6

Using the Order of Operations Use the order of operations to evaluate each of the following expressions.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Solution

ⓐ

ⓑ

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.
ⓒ

14 1 • Prerequisites
ⓓ
In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.
ⓔ

TRY IT #6 Use the order of operations to evaluate each of the following expressions.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Using Properties of Real Numbers
For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.
Commutative Properties The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.
Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example, Similarly,

is not the same as

Associative Properties The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

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Are subtraction and division associative? Review these examples.

1.1 • Real Numbers: Algebra Essentials 15

As we can see, neither subtraction nor division is associative. Distributive Property The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.
This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.
Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products. To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference

We can rewrite the difference of the two terms 12 and

by

turning the subtraction expression into addition of the opposite. So instead of subtracting

we add the opposite.

Now, distribute and simplify the result.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

Identity Properties The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

For example, we have

and

real number, including 0 and 1.

There are no exceptions for these properties; they work for every

16 1 • Prerequisites

Inverse Properties The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted by (−a), that, when added to the original number, results in the additive identity, 0.

For example, if

the additive inverse is 8, since

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if

the reciprocal, denoted is because

Properties of Real Numbers

The following properties hold for real numbers a, b, and c.

Addition

Multiplication

Commutative Property

Associative Property

Distributive Property

Identity Property

There exists a unique real number called the additive identity, 0, such that, for any real number a

There exists a unique real number called the multiplicative identity, 1, such that, for any real
number a

Inverse Property

Every real number a has an additive inverse, or opposite, denoted –a, such that

Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted
such that

EXAMPLE 7

Using Properties of Real Numbers Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

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Solution
ⓐ
ⓑ
ⓒ
ⓓ

1.1 • Real Numbers: Algebra Essentials 17

ⓔ

TRY IT #7

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Evaluating Algebraic Expressions
So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see

expressions such as

or

In the expression

5 is called a constant because it does not vary

and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.
Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

18 1 • Prerequisites

EXAMPLE 8

Describing Algebraic Expressions List the constants and variables for each algebraic expression.

ⓐ x+5 ⓑ

ⓒ

Solution

Constants Variables

a. x + 5

5

x

b.

c.

2

TRY IT #8

List the constants and variables for each algebraic expression.

ⓐ

ⓑ 2(L + W) ⓒ

EXAMPLE 9

Evaluating an Algebraic Expression at Different Values

Evaluate the expression

for each value for x.

ⓐ

ⓑ

Solution

ⓐ Substitute 0 for

ⓒ

ⓓ

ⓑ Substitute 1 for

ⓒ Substitute for

ⓓ Substitute for

TRY IT #9

Evaluate the expression

ⓐ

ⓑ

for each value for y.

ⓒ

ⓓ

EXAMPLE 10

Evaluating Algebraic Expressions Evaluate each expression for the given values.

ⓐ

for

ⓑ

for

ⓒ

for

ⓓ

for

ⓔ

for

Solution

ⓐ Substitute for

ⓑ Substitute 10 for

ⓒ Substitute 5 for

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ⓓ Substitute 11 for and –8 for

ⓔ Substitute 2 for and 3 for

1.1 • Real Numbers: Algebra Essentials 19

TRY IT #10

Evaluate each expression for the given values.

ⓐ

for

ⓑ

for

ⓒ

ⓓ

for

ⓔ

for for

Formulas
An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation has the solution of 3 because when we substitute 3 for in the equation, we obtain the true statement
A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area of a circle in terms of the radius of the circle: For any value of the area can be found by evaluating the expression

EXAMPLE 11

Using a Formula A right circular cylinder with radius and height has the surface area (in square units) given by the formula
See Figure 3. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of

Solution Evaluate the expression

Figure 3 Right circular cylinder

for

and

The surface area is

square inches.

TRY IT #11

A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area

of the mat (in square centimeters, or cm2) is found to be

See

Figure 4. Find the area of a mat for a photograph with length 32 cm and width 24 cm.

20 1 • Prerequisites

Figure 4

Simplifying Algebraic Expressions
Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

EXAMPLE 12

Simplifying Algebraic Expressions Simplify each algebraic expression.

ⓐ

ⓑ

ⓒ

ⓓ

Solution

ⓐ

ⓑ

ⓒ

ⓓ

TRY IT #12 Simplify each algebraic expression.

ⓐ

ⓑ

ⓒ

ⓓ

EXAMPLE 13
Simplifying a Formula A rectangle with length and width has a perimeter given by

Simplify this expression.

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Solution

1.1 • Real Numbers: Algebra Essentials 21

TRY IT #13

If the amount is deposited into an account paying simple interest for time the total value of

the deposit is given by

Simplify the expression. (This formula will be explored in

more detail later in the course.)

MEDIA
Access these online resources for additional instruction and practice with real numbers.
Simplify an Expression. (http://openstax.org/l/simexpress) Evaluate an Expression 1. (http://openstax.org/l/ordofoper1) Evaluate an Expression 2. (http://openstax.org/l/ordofoper2)

1.1 SECTION EXERCISES
Verbal

1. Is an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

2. What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

3. What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

Numeric

For the following exercises, simplify the given expression.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

22 1 • Prerequisites

Algebraic

For the following exercises, evaluate the expressions using the given variable.

28.

for

29.

for

30.

for

31.

for

32.

for

33.

for

34. For the

35.

for

37.

for

for

36.

for

For the following exercises, simplify the expression.

38.

39.

41.

42.

44.

45.

47.

48.

50.

51.

40. 43. 46. 49.
52.

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

53. Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

54. How much money does Fred keep?

For the following exercises, solve the given problem.
55. According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by Is the circumference of a quarter a whole number, a rational number, or an irrational number?

56. Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

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1.1 • Real Numbers: Algebra Essentials 23

For the following exercises, consider this scenario: There is a mound of pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

57. Write the equation that describes the situation.

58. Solve for g.

For the following exercise, solve the given problem.

59. Ramon runs the marketing department at their

company. Their department gets a budget every

year, and every year, they must spend the entire

budget without going over. If they spend less than

the budget, then the department gets a smaller

budget the following year. At the beginning of this

year, Ramon got $2.5 million for the annual

marketing budget. They must spend the budget

such that

What property of

addition tells us what the value of x must be?

Technology

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

60.

61.

Extensions

62. If a whole number is not a natural number, what must the number be?

63. Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

64. Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

65. Determine whether the simplified expression is rational or irrational:

66. Determine whether the simplified expression is rational or irrational:

67. The division of two natural numbers will always result in what type of number?

68. What property of real numbers would simplify the following expression:

24 1 • Prerequisites

1.2 Exponents and Scientific Notation
Learning Objectives
In this section, you will: Use the product rule of exponents. Use the quotient rule of exponents. Use the power rule of exponents. Use the zero exponent rule of exponents. Use the negative rule of exponents. Find the power of a product and a quotient. Simplify exponential expressions. Use scientific notation.

Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number.

Using a calculator, we enter

and press ENTER. The calculator displays 1.304596316E13.

What does this mean? The “E13” portion of the result represents the exponent 13 of ten, so there are a maximum of

approximately

bits of data in that one-hour film. In this section, we review rules of exponents first and then

apply them to calculations involving very large or small numbers.

Using the Product Rule of Exponents

Consider the product

Both terms have the same base, x, but they are raised to different exponents. Expand each

expression, and then rewrite the resulting expression.

The result is that
Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.

Now consider an example with real numbers.

We can always check that this is true by simplifying each exponential expression. We find that is 8, is 16, and is

128. The product

equals 128, so the relationship is true. We can use the product rule of exponents to simplify

expressions that are a product of two numbers or expressions with the same base but different exponents.

The Product Rule of Exponents

For any real number and natural numbers and the product rule of exponents states that

EXAMPLE 1
Using the Product Rule Write each of the following products with a single base. Do not simplify further.

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1.2 • Exponents and Scientific Notation 25

ⓐ

ⓑ

ⓒ

Solution

Use the product rule to simplify each expression.

ⓐ

ⓑ

ⓒ

At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of

multiplication, begin by simplifying the first two.

Notice we get the same result by adding the three exponents in one step.

TRY IT #1 Write each of the following products with a single base. Do not simplify further.

ⓐ

ⓑ

ⓒ

Using the Quotient Rule of Exponents
The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as where

Consider the example Perform the division by canceling common factors.

Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.
In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.

For the time being, we must be aware of the condition

Otherwise, the difference

could be zero or negative.

Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify

matters and assume from here on that all variables represent nonzero real numbers.

The Quotient Rule of Exponents

For any real number and natural numbers and such that

the quotient rule of exponents states that

EXAMPLE 2
Using the Quotient Rule Write each of the following products with a single base. Do not simplify further.

26 1 • Prerequisites

ⓐ

ⓑ

ⓒ

Solution Use the quotient rule to simplify each expression.

ⓐ

ⓑ

ⓒ

TRY IT #2 Write each of the following products with a single base. Do not simplify further.

ⓐ

ⓑ

ⓒ

Using the Power Rule of Exponents

Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the

power rule of exponents. Consider the expression

The expression inside the parentheses is multiplied twice

because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3.

The exponent of the answer is the product of the exponents:

In other words, when raising an

exponential expression to a power, we write the result with the common base and the product of the exponents.

Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents.

The Power Rule of Exponents For any real number and positive integers and the power rule of exponents states that

EXAMPLE 3

Using the Power Rule Write each of the following products with a single base. Do not simplify further.

ⓐ

ⓑ

ⓒ

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1.2 • Exponents and Scientific Notation 27

Solution Use the power rule to simplify each expression.

ⓐ

ⓑ

ⓒ

TRY IT #3 Write each of the following products with a single base. Do not simplify further.

ⓐ

ⓑ

ⓒ

Using the Zero Exponent Rule of Exponents

Return to the quotient rule. We made the condition that

so that the difference

would never be zero or

negative. What would happen if

In this case, we would use the zero exponent rule of exponents to simplify the

expression to 1. To see how this is done, let us begin with an example.

If we were to simplify the original expression using the quotient rule, we would have

If we equate the two answers, the result is a real number.

This is true for any nonzero real number, or any variable representing

The sole exception is the expression value to be undefined.

This appears later in more advanced courses, but for now, we will consider the

The Zero Exponent Rule of Exponents

For any nonzero real number the zero exponent rule of exponents states that

EXAMPLE 4
Using the Zero Exponent Rule Simplify each expression using the zero exponent rule of exponents.

ⓐⓑ

ⓒ

ⓓ

Solution Use the zero exponent and other rules to simplify each expression.

ⓐ

ⓑ

28 1 • Prerequisites
ⓒ

ⓓ

TRY IT #4 Simplify each expression using the zero exponent rule of exponents.

ⓐⓑ

ⓒ

ⓓ

Using the Negative Rule of Exponents

Another useful result occurs if we relax the condition that

simplify When

—that is, where the difference

simplify the expression to its reciprocal.

in the quotient rule even further. For example, can we is negative—we can use the negative rule of exponents to

Divide one exponential expression by another with a larger exponent. Use our example,

If we were to simplify the original expression using the quotient rule, we would have

Putting the answers together, we have a nonzero real number.
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This is true for any nonzero real number, or any variable representing

1.2 • Exponents and Scientific Notation 29
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
We have shown that the exponential expression is defined when is a natural number, 0, or the negative of a natural number. That means that is defined for any integer Also, the product and quotient rules and all of the rules we will look at soon hold for any integer
The Negative Rule of Exponents For any nonzero real number and natural number the negative rule of exponents states that

EXAMPLE 5
Using the Negative Exponent Rule Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

ⓐ

ⓑ

Solution
ⓐ

ⓒ

ⓒ ⓑ

TRY IT #5

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

ⓐ

ⓑ

ⓒ

EXAMPLE 6

Using the Product and Quotient Rules Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

ⓐ

ⓑ

ⓒ

Solution

ⓐ

ⓑ

ⓒ

TRY IT #6

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

ⓐ

ⓑ

Finding the Power of a Product
To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider
We begin by using the associative and commutative properties of multiplication to regroup the factors.

30 1 • Prerequisites

In other words, The Power of a Product Rule of Exponents For any real numbers and and any integer the power of a product rule of exponents states that

EXAMPLE 7

Using the Power of a Product Rule Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Solution

Use the product and quotient rules and the new definitions to simplify each expression.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

TRY IT #7

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Finding the Power of a Quotient
To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, let’s look at the following example.

Let’s rewrite the original problem differently and look at the result.

It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.

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1.2 • Exponents and Scientific Notation 31

The Power of a Quotient Rule of Exponents For any real numbers and and any integer the power of a quotient rule of exponents states that

EXAMPLE 8
Using the Power of a Quotient Rule Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Solution

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

TRY IT #8

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Simplifying Exponential Expressions
Recall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the expression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.

EXAMPLE 9

Simplifying Exponential Expressions Simplify each expression and write the answer with positive exponents only.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

ⓕ

32 1 • Prerequisites
Solution
ⓐ
ⓑ ⓒ
ⓓ ⓔ ⓕ

TRY IT #9 Simplify each expression and write the answer with positive exponents only.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

ⓕ

Using Scientific Notation

Recall at the beginning of the section that we found the number

when describing bits of information in digital

images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an

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1.2 • Exponents and Scientific Notation 33
electron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers such as these?
A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places n that you moved the decimal point. Multiply the decimal number by 10 raised to a power of n. If you moved the decimal left as in a very large number, is positive. If you moved the decimal right as in a small large number, is negative.
For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.

We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.
Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.

Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

Scientific Notation A number is written in scientific notation if it is written in the form

where

and is an integer.

EXAMPLE 10
Converting Standard Notation to Scientific Notation Write each number in scientific notation.
ⓐ Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m ⓑ Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m ⓒ Number of stars in Andromeda Galaxy: 1,000,000,000,000 ⓓ Diameter of electron: 0.00000000000094 m ⓔ Probability of being struck by lightning in any single year: 0.00000143
Solution
ⓐ

ⓑ

34 1 • Prerequisites

ⓒ

ⓓ

ⓔ

Analysis Observe that, if the given number is greater than 1, as in examples a–c, the exponent of 10 is positive; and if the number is less than 1, as in examples d–e, the exponent is negative.

TRY IT #10

Write each number in scientific notation.
ⓐ U.S. national debt per taxpayer (April 2014): $152,000 ⓑ World population (April 2014): 7,158,000,000 ⓒ World gross national income (April 2014): $85,500,000,000,000 ⓓ Time for light to travel 1 m: 0.00000000334 s ⓔ Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715

Converting from Scientific to Standard Notation
To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal places to the right if is positive or places to the left if is negative and add zeros as needed. Remember, if is positive, the value of the number is greater than 1, and if is negative, the value of the number is less than one.

EXAMPLE 11

Converting Scientific Notation to Standard Notation Convert each number in scientific notation to standard notation.

ⓐ

ⓑ

ⓒ

ⓓ

Solution

ⓐ

ⓑ

ⓒ

ⓓ

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1.2 • Exponents and Scientific Notation 35

TRY IT #11

Convert each number in scientific notation to standard notation.

ⓐ

ⓑ

ⓒ

ⓓ

Using Scientific Notation in Applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than

doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water.

Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around

molecules of water and 1 L of water holds about

average drops. Therefore, there are

approximately

atoms in 1 L of water. We simply multiply the decimal

terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For

example, consider the product

The answer is not in proper scientific notation

because 35 is greater than 10. Consider 35 as

That adds a ten to the exponent of the answer.

EXAMPLE 12

Using Scientific Notation Perform the operations and write the answer in scientific notation.

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

Solution

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

TRY IT #12 Perform the operations and write the answer in scientific notation.

36 1 • Prerequisites

ⓐ

ⓑ

ⓒ

ⓓ

ⓔ

EXAMPLE 13
Applying Scientific Notation to Solve Problems In April 2014, the population of the United States was about 308,000,000 people. The national debt was about $17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the amount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.
Solution The population was
The national debt was
To find the amount of debt per citizen, divide the national debt by the number of citizens.

The debt per citizen at the time was about

or $57,000.

TRY IT #13

An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures approximately 0.000008 m long. Write each number in scientific notation and find the total length if the cells were laid end-to-end. Write the answer in both scientific and standard notations.

MEDIA
Access these online resources for additional instruction and practice with exponents and scientific notation.
Exponential Notation (http://openstax.org/l/exponnot) Properties of Exponents (http://openstax.org/l/exponprops) Zero Exponent (http://openstax.org/l/zeroexponent) Simplify Exponent Expressions (http://openstax.org/l/exponexpres) Quotient Rule for Exponents (http://openstax.org/l/quotofexpon) Scientific Notation (http://openstax.org/l/scientificnota) Converting to Decimal Notation (http://openstax.org/l/decimalnota)

1.2 SECTION EXERCISES
Verbal

1. Is the same as Explain.

2. When can you add two exponents?

4. Explain what a negative exponent does.

3. What is the purpose of scientific notation?

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1.2 • Exponents and Scientific Notation 37

Numeric

For the following exercises, simplify the given expression. Write answers with positive exponents.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

15.

16.

17.

18.

19.

20.

For the following exercises, express the decimal in scientific notation.

21. 0.0000314

22. 148,000,000

For the following exercises, convert each number in scientific notation to standard notation.

23.

24.

Algebraic

For the following exercises, simplify the given expression. Write answers with positive exponents.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

38 1 • Prerequisites
43.

Real-World Applications

44. To reach escape velocity, a

rocket must travel at the

rate of

ft/min.

Rewrite the rate in

standard notation.

45. A dime is the thinnest coin in U.S. currency. A dime’s thickness measures m. Rewrite the number in standard notation.

46. The average distance between Earth and the Sun is 92,960,000 mi. Rewrite the distance using scientific notation.

47. A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

48. The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was Rewrite the GDP in standard notation.

49. One picometer is approximately in. Rewrite this length using standard notation.

50. The value of the services sector of the U.S. economy in the first quarter of 2012 was $10,633.6 billion. Rewrite this amount in scientific notation.

Technology

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

51.

52.

Extensions

For the following exercises, simplify the given expression. Write answers with positive exponents.

53.

54.

55.

56.

57.

58. Avogadro’s constant is

used to calculate the

number of particles in a

mole. A mole is a basic unit

in chemistry to measure

the amount of a substance.

The constant is

Write

Avogadro’s constant in

standard notation.

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59. Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as Write Planck’s constant in standard notation.

1.3 • Radicals and Rational Exponents 39

1.3 Radicals and Rational Exponents
Learning Objectives
In this section, you will: Evaluate square roots. Use the product rule to simplify square roots. Use the quotient rule to simplify square roots. Add and subtract square roots. Rationalize denominators. Use rational roots.
A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in Figure 1, and use the Pythagorean Theorem.

Figure 1

Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.

Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Since

the square root of is

The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo

squaring, we take the square root.

In general terms, if is a positive real number, then the square root of is a number that, when multiplied by itself, gives The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals The square root obtained using a calculator is the principal square root.

The principal square root of is written as The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.

40 1 • Prerequisites

Principal Square Root
The principal square root of is the nonnegative number that, when multiplied by itself, equals It is written as a radical expression, with a symbol called a radical over the term called the radicand:

Q&A

Does

No. Although both and

are the radical symbol implies only a nonnegative root, the principal

square root. The principal square root of 25 is

EXAMPLE 1

Evaluating Square Roots Evaluate each expression.

ⓐ

ⓑ

ⓒ

ⓓ

Solution

ⓐ

because

ⓑ

because

and

ⓒ

because

ⓓ

because

and

Q&A

For

can we find the square roots before adding?

No.

This is not equivalent to

The order of operations

requires us to add the terms in the radicand before finding the square root.

TRY IT #1 Evaluate each expression.

ⓐ

ⓑ

ⓒ

ⓓ

Using the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several

properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into

the product of two separate rational expressions. For instance, we can rewrite

as

We can also use the

product rule to express the product of multiple radical expressions as a single radical expression.

The Product Rule for Simplifying Square Roots

If and are nonnegative, the square root of the product is equal to the product of the square roots of and

...
HOW TO Given a square root radical expression, use the product rule to simplify it.
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1. Factor any perfect squares from the radicand. 2. Write the radical expression as a product of radical expressions. 3. Simplify.

EXAMPLE 2

Using the Product Rule to Simplify Square Roots Simplify the radical expression.

ⓐ

ⓑ

Solution

ⓐ

ⓑ

1.3 • Radicals and Rational Exponents 41

TRY IT #2 Simplify
...
HOW TO
Given the product of multiple radical expressions, use the product rule to combine them into one radical expression. 1. Express the product of multiple radical expressions as a single radical expression. 2. Simplify.
EXAMPLE 3 Using the Product Rule to Simplify the Product of Multiple Square Roots Simplify the radical expression.
Solution

TRY IT #3 Simplify

assuming

Using the Quotient Rule to Simplify Square Roots
Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately.

42 1 • Prerequisites

We can rewrite

as

The Quotient Rule for Simplifying Square Roots The square root of the quotient is equal to the quotient of the square roots of and where

...
HOW TO
Given a radical expression, use the quotient rule to simplify it. 1. Write the radical expression as the quotient of two radical expressions. 2. Simplify the numerator and denominator.
EXAMPLE 4 Using the Quotient Rule to Simplify Square Roots Simplify the radical expression.
Solution

TRY IT #4 Simplify
EXAMPLE 5 Using the Quotient Rule to Simplify an Expression with Two Square Roots Simplify the radical expression.
Solution

TRY IT #5 Simplify
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1.3 • Radicals and Rational Exponents 43

Adding and Subtracting Square Roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical

type such as square roots. For example, the sum of and

is

However, it is often possible to simplify radical

expressions, and that may change the radicand. The radical expression

can be written with a in the radicand, as

so

...
HOW TO

Given a radical expression requiring addition or subtraction of square roots, simplify.
1. Simplify each radical expression. 2. Add or subtract expressions with equal radicands.

EXAMPLE 6

Adding Square Roots Add

Solution

We can rewrite

as

expression becomes

According the product rule, this becomes

The square root of

which is

Now the terms have the same radicand so we can add.

is 2, so the

TRY IT #6 Add
EXAMPLE 7 Subtracting Square Roots Subtract
Solution Rewrite each term so they have equal radicands.

Now the terms have the same radicand so we can subtract. TRY IT #7 Subtract

44 1 • Prerequisites

Rationalizing Denominators

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.

We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.

For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the

denominator is

multiply by

For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and

denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the

denominator. If the denominator is

then the conjugate is

...
HOW TO

Given an expression with a single square root radical term in the denominator, rationalize the denominator.
a. Multiply the numerator and denominator by the radical in the denominator. b. Simplify.

EXAMPLE 8

Rationalizing a Denominator Containing a Single Term

Write

in simplest form.

Solution The radical in the denominator is

So multiply the fraction by

Then simplify.

TRY IT #8 Write

in simplest form.

...
HOW TO
Given an expression with a radical term and a constant in the denominator, rationalize the denominator.
1. Find the conjugate of the denominator. 2. Multiply the numerator and denominator by the conjugate. 3. Use the distributive property. 4. Simplify.

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1.3 • Radicals and Rational Exponents 45

EXAMPLE 9

Rationalizing a Denominator Containing Two Terms

Write

in simplest form.

Solution Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate

of

is

Then multiply the fraction by

TRY IT #9 Write

in simplest form.

Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.

Understanding nth Roots

Suppose we know that

We want to find what number raised to the 3rd power is equal to 8. Since

that 2 is the cube root of 8.

we say

The nth root of is a number that, when raised to the nth power, gives For example, is the 5th root of

because

If is a real number with at least one nth root, then the principal nth root of is the number

with the same sign as that, when raised to the nth power, equals

The principal nth root of is written as where is a positive integer greater than or equal to 2. In the radical expression, is called the index of the radical.

Principal th Root

If is a real number with at least one nth root, then the principal nth root of written as the same sign as that, when raised to the nth power, equals The index of the radical is

is the number with

EXAMPLE 10

Simplifying nth Roots Simplify each of the following:

ⓐ

ⓑ

Solution

ⓒ

ⓓ

ⓐ

because

ⓑ First, express the product as a single radical expression.

ⓒ

because

46 1 • Prerequisites

ⓓ

TRY IT #10 Simplify.

ⓐ

ⓑ

ⓒ

Using Rational Exponents
Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index is even, then cannot be negative.

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.
Rational Exponents
Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

...
HOW TO
Given an expression with a rational exponent, write the expression as a radical.
1. Determine the power by looking at the numerator of the exponent. 2. Determine the root by looking at the denominator of the exponent. 3. Using the base as the radicand, raise the radicand to the power and use the root as the index.

EXAMPLE 11

Writing Rational Exponents as Radicals

Write

as a radical. Simplify.

Solution The 2 tells us the power and the 3 tells us the root.

We know that

because

Because the cube root is easy to find, it is easiest to find the cube root

before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

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1.3 • Radicals and Rational Exponents 47

TRY IT #11 Write as a radical. Simplify.

EXAMPLE 12

Writing Radicals as Rational Exponents

Write

using a rational exponent.

Solution The power is 2 and the root is 7, so the rational exponent will be We get

Using properties of exponents, we get

TRY IT #12 Write

using a rational exponent.

EXAMPLE 13

Simplifying Rational Exponents Simplify:

ⓐ

ⓑ

Solution
ⓐ

ⓑ
           
TRY IT #13 Simplify
MEDIA Access these online resources for additional instruction and practice with radicals and rational exponents. Radicals (http://openstax.org/l/introradical) Rational Exponents (http://openstax.org/l/rationexpon) Simplify Radicals (http://openstax.org/l/simpradical) Rationalize Denominator (http://openstax.org/l/rationdenom)

48 1 • Prerequisites

1.3 SECTION EXERCISES
Verbal

1. What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.

2. Where would radicals come in the order of operations? Explain why.

3. Every number will have two square roots. What is the principal square root?

4. Can a radical with a negative radicand have a real square root? Why or why not?

Numeric

For the following exercises, simplify each expression.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

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Algebraic

For the following exercises, simplify each expression.

35.

36.

38.

39.

41.

42.

44.

45.

47.

48.

50.

51.

53.

54.

56.

57.

1.3 • Radicals and Rational Exponents 49
37. 40. 43. 46. 49. 52. 55. 58.

59.

60.

61.

62.

63.

64.

Real-World Applications

65. A guy wire for a suspension bridge runs from the

ground diagonally to the top of the closest pylon

to make a triangle. We can use the Pythagorean

Theorem to find the length of guy wire needed.

The square of the distance between the wire on

the ground and the pylon on the ground is 90,000

feet. The square of the height of the pylon is

160,000 feet. So the length of the guy wire can be

found by evaluating

What is

the length of the guy wire?

66. A car accelerates at a rate of

where

t is the time in seconds after the car moves from rest. Simplify the expression.

50 1 • Prerequisites

Extensions
For the following exercises, simplify each expression.

67.

68.

69.

70.

71.

72.

73.

1.4 Polynomials
Learning Objectives
In this section, you will: Identify the degree and leading coefficient of polynomials. Add and subtract polynomials. Multiply polynomials. Use FOIL to multiply binomials. Perform operations with polynomials of several variables.
Maahi is building a little free library (a small house-shaped book repository), whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which people can take and donate books. Maahi wants to find the area of the front of the library so that they can purchase the correct amount of paint. Using the measurements of the front of the house, shown in Figure 1, we can create an expression that combines several variable terms, allowing us to solve this problem and others like it.

First find the area of the square in square feet.

Figure 1

Then find the area of the triangle in square feet.

Next find the area of the rectangular door in square feet.

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1.4 • Polynomials 51

The area of the front of the library can be found by adding the areas of the square and the triangle, and then subtracting

the area of the rectangle. When we do this, we get

or

ft2.

In this section, we will examine expressions such as this one, which combine several variable terms.

Identifying the Degree and Leading Coefficient of Polynomials

The formula just found is an example of a polynomial, which is a sum of or difference of terms, each consisting of a

variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as

is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or

fractions. Each product

such as

is a term of a polynomial. If a term does not contain a variable, it is called

a constant.

A polynomial containing only one term, such as

is called a monomial. A polynomial containing two terms, such as

is called a binomial. A polynomial containing three terms, such as

is called a trinomial.

We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.

Polynomials A polynomial is an expression that can be written in the form

Each real number ai is called a coefficient. The number that is not multiplied by a variable is called a constant.

Each product

is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called

the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the

leading coefficient.

...
HOW TO
Given a polynomial expression, identify the degree and leading coefficient.
1. Find the highest power of x to determine the degree. 2. Identify the term containing the highest power of x to find the leading term. 3. Identify the coefficient of the leading term.

EXAMPLE 1

Identifying the Degree and Leading Coefficient of a Polynomial For the following polynomials, identify the degree, the leading term, and the leading coefficient.

ⓐ

ⓑ

ⓒ

Solution

ⓐ The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree,
leading coefficient is the coefficient of that term,

ⓑ The highest power of t is so the degree is The leading term is the term containing that degree,
leading coefficient is the coefficient of that term,

The The

52 1 • Prerequisites

ⓒ The highest power of p is so the degree is The leading term is the term containing that degree,

The

leading coefficient is the coefficient of that term,

TRY IT #1 Identify the degree, leading term, and leading coefficient of the polynomial

Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to

the same exponents. For example, and

are like terms, and can be added to get

but and are not

like terms, and therefore cannot be added.

...
HOW TO

Given multiple polynomials, add or subtract them to simplify the expressions.
1. Combine like terms. 2. Simplify and write in standard form.

EXAMPLE 2
Adding Polynomials Find the sum.
Solution
     
Analysis We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.
TRY IT #2 Find the sum.

EXAMPLE 3 Subtracting Polynomials Find the difference.
Solution

Analysis Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial
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to the first. TRY IT #3 Find the difference.

1.4 • Polynomials 53

Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of

the polynomial. We can distribute the in

to obtain the equivalent expression

When multiplying

polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second.

We then add the products together and combine like terms to simplify.

...
HOW TO

Given the multiplication of two polynomials, use the distributive property to simplify the expression.
1. Multiply each term of the first polynomial by each term of the second. 2. Combine like terms. 3. Simplify.

EXAMPLE 4
Multiplying Polynomials Using the Distributive Property Find the product.
Solution
           
Analysis We can use a table to keep track of our work, as shown in Table 1. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.

Table 1

54 1 • Prerequisites
Table 1
TRY IT #4 Find the product.
Using FOIL to Multiply Binomials
A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial.
The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.
...
HOW TO Given two binomials, use FOIL to simplify the expression. 1. Multiply the first terms of each binomial. 2. Multiply the outer terms of the binomials. 3. Multiply the inner terms of the binomials. 4. Multiply the last terms of each binomial. 5. Add the products. 6. Combine like terms and simplify.
EXAMPLE 5 Using FOIL to Multiply Binomials Use FOIL to find the product.
Solution Find the product of the first terms.
Find the product of the outer terms.
Find the product of the inner terms.
Find the product of the last terms.
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1.4 • Polynomials 55
TRY IT #5 Use FOIL to find the product.
Perfect Square Trinomials
Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.
Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.
Perfect Square Trinomials When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.
...
HOW TO Given a binomial, square it using the formula for perfect square trinomials. 1. Square the first term of the binomial. 2. Square the last term of the binomial. 3. For the middle term of the trinomial, double the product of the two terms. 4. Add and simplify.
EXAMPLE 6 Expanding Perfect Squares Expand
Solution Begin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two terms.
Simplify.

56 1 • Prerequisites

TRY IT #6 Expand

Difference of Squares

Another special product is called the difference of squares, which occurs when we multiply a binomial by another

binomial with the same terms but the opposite sign. Let’s see what happens when we multiply

using the

FOIL method.

The middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, let’s look at a few examples.

Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.
Q&A Is there a special form for the sum of squares? No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.
Difference of Squares
When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.

...
HOW TO
Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares.
1. Square the first term of the binomials. 2. Square the last term of the binomials. 3. Subtract the square of the last term from the square of the first term.

EXAMPLE 7

Multiplying Binomials Resulting in a Difference of Squares Multiply

Solution

Square the first term to get

Square the last term to get

the square of the first term to find the product of

Subtract the square of the last term from

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1.4 • Polynomials 57
TRY IT #7 Multiply
Performing Operations with Polynomials of Several Variables
We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

EXAMPLE 8
Multiplying Polynomials Containing Several Variables Multiply
Solution Follow the same steps that we used to multiply polynomials containing only one variable.

TRY IT #8 Multiply
MEDIA Access these online resources for additional instruction and practice with polynomials. Adding and Subtracting Polynomials (http://openstax.org/l/addsubpoly) Multiplying Polynomials (http://openstax.org/l/multiplpoly) Special Products of Polynomials (http://openstax.org/l/specialpolyprod)

1.4 SECTION EXERCISES
Verbal

1. Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.

2. Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.

3. You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?

4. State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.

58 1 • Prerequisites

Algebraic

For the following exercises, identify the degree of the polynomial.

5.

6.

7.

8.

9.

10.

For the following exercises, find the sum or difference.

11.

12.

13.

14.

15.

16.

For the following exercises, find the product.

17.

18.

19.

20.

21.

22.

23.

For the following exercises, expand the binomial.

24.

25.

26.

27.

28.

29.

30.

For the following exercises, multiply the binomials.

31.

32.

33.

34.

35.

36.

37.

For the following exercises, multiply the polynomials.

38.

39.

40.

41.

42.

43.

44.

45.

46.

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1.5 • Factoring Polynomials 59

47.

48.

49.

50.

51.

52.

Real-World Applications
53. A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: where x is measured in meters. Multiply the binomials to find the area of the plot in standard form.

54. A prospective buyer wants to know how much

grain a specific silo can hold. The area of the floor

of the silo is

The height of the silo is

where x is measured in feet. Expand the

square and multiply by the height to find the

expression that shows how much grain the silo

can hold.

Extensions

For the following exercises, perform the given operations.

55.

56.

57.

1.5 Factoring Polynomials
Learning Objectives
In this section, you will: Factor the greatest common factor of a polynomial. Factor a trinomial. Factor by grouping. Factor a perfect square trinomial. Factor a difference of squares. Factor the sum and difference of cubes. Factor expressions using fractional or negative exponents.
Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in Figure 1.

Figure 1 The area of the entire region can be found using the formula for the area of a rectangle.
The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The

60 1 • Prerequisites

two square regions each have an area of

and one side of length giving an area of

subtracted has an area of

units2.

units2. The other rectangular region has one side of length units2. So the region that must be

The area of the region that requires grass seed is found by subtracting

units2. This area can also be

expressed in factored form as

units2. We can confirm that this is an equivalent expression by multiplying.

Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

Factoring the Greatest Common Factor of a Polynomial

When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that

divides evenly into both numbers. For instance, is the GCF of and because it is the largest number that divides

evenly into both and The GCF of polynomials works the same way: is the GCF of and

because it is the

largest polynomial that divides evenly into both and

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

Greatest Common Factor

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

...
HOW TO
Given a polynomial expression, factor out the greatest common factor.
1. Identify the GCF of the coefficients. 2. Identify the GCF of the variables. 3. Combine to find the GCF of the expression. 4. Determine what the GCF needs to be multiplied by to obtain each term in the expression. 5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

EXAMPLE 1

Factoring the Greatest Common Factor Factor

Solution

First, find the GCF of the expression. The GCF of 6, 45, and 21 is 3. The GCF of

, and is . (Note that the GCF of a

set of expressions in the form will always be the exponent of lowest degree.) And the GCF of

, and is .

Combine these to find the GCF of the polynomial, .

Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that , and

Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.

Analysis After factoring, we can check our work by multiplying. Use the distributive property to confirm that

TRY IT #1 Factor

by pulling out the GCF.

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1.5 • Factoring Polynomials 61

Factoring a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial

expressions can be factored. The polynomial

has a GCF of 1, but it can be written as the product of the

factors

and

Trinomials of the form trinomial is and their sum is

can be factored by finding two numbers with a product of and a sum of The

for example, can be factored using the numbers and because the product of those numbers

The trinomial can be rewritten as the product of

and

Factoring a Trinomial with Leading Coefficient 1

A trinomial of the form

can be written in factored form as

where

and

Q&A

Can every trinomial be factored as a product of binomials? No. Some polynomials cannot be factored. These polynomials are said to be prime.

...
HOW TO

Given a trinomial in the form

factor it.

1. List factors of 2. Find and a pair of factors of with a sum of 3. Write the factored expression

EXAMPLE 2

Factoring a Trinomial with Leading Coefficient 1 Factor

Solution

We have a trinomial with leading coefficient

and

We need to find two numbers with a product of

and a sum of In the table below, we list factors until we find a pair with the desired sum.

Factors of

Sum of Factors

14

2

Now that we have identified and as and write the factored form as
Analysis We can check our work by multiplying. Use FOIL to confirm that

Q&A

Does the order of the factors matter? No. Multiplication is commutative, so the order of the factors does not matter.

62 1 • Prerequisites

TRY IT #2 Factor

Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can

factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression

separately, and then factoring out the GCF of the entire expression. The trinomial

can be rewritten as

using this process. We begin by rewriting the original expression as

and then factor

each portion of the expression to obtain

We then pull out the GCF of

to find the factored

expression.

Factor by Grouping

To factor a trinomial in the form

by grouping, we find two numbers with a product of and a sum of

We use these numbers to divide the term into the sum of two terms and factor each portion of the expression

separately, then factor out the GCF of the entire expression.

...
HOW TO

Given a trinomial in the form

factor by grouping.

1. List factors of

2. Find and a pair of factors of with a sum of

3. Rewrite the original expression as

4. Pull out the GCF of

5. Pull out the GCF of

6. Factor out the GCF of the expression.

EXAMPLE 3

Factoring a Trinomial by Grouping

Factor

by grouping.

Solution We have a trinomial with product of and a sum of

and

First, determine

We need to find two numbers with a

In the table below, we list factors until we find a pair with the desired sum.

Factors of

Sum of Factors

29

13

7

So

and

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1.5 • Factoring Polynomials 63

Analysis We can check our work by multiplying. Use FOIL to confirm that

TRY IT #3 Factor

ⓐ

ⓑ

Factoring a Perfect Square Trinomial
A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

We can use this equation to factor any perfect square trinomial. Perfect Square Trinomials A perfect square trinomial can be written as the square of a binomial:

...
HOW TO
Given a perfect square trinomial, factor it into the square of a binomial.
1. Confirm that the first and last term are perfect squares. 2. Confirm that the middle term is twice the product of 3. Write the factored form as

EXAMPLE 4

Factoring a Perfect Square Trinomial Factor

Solution

Notice that

and are perfect squares because

and

twice the product of and The middle term is, indeed, twice the product:

a perfect square trinomial and can be written as

Then check to see if the middle term is Therefore, the trinomial is

TRY IT #4 Factor
Factoring a Difference of Squares
A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when

64 1 • Prerequisites
the two factors are multiplied.
We can use this equation to factor any differences of squares. Differences of Squares A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

...
HOW TO
Given a difference of squares, factor it into binomials. 1. Confirm that the first and last term are perfect squares. 2. Write the factored form as

EXAMPLE 5
Factoring a Difference of Squares Factor
Solution Notice that and are perfect squares because squares and can be rewritten as

and

The polynomial represents a difference of

TRY IT #5 Factor

Q&A

Is there a formula to factor the sum of squares? No. A sum of squares cannot be factored.

Factoring the Sum and Difference of Cubes
Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

Similarly, the difference of cubes can be factored into a binomial and a trinomial, but with different signs.

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.

The sign of the first 2 is the same as the sign between And the sign of the last term, 4, is always positive.

The sign of the term is opposite the sign between

Sum and Difference of Cubes

We can factor the sum of two cubes as

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1.5 • Factoring Polynomials 65

We can factor the difference of two cubes as

...
HOW TO

Given a sum of cubes or difference of cubes, factor it.

1. Confirm that the first and last term are cubes,

or

2. For a sum of cubes, write the factored form as

factored form as

For a difference of cubes, write the

EXAMPLE 6

Factoring a Sum of Cubes Factor

Solution Notice that and

are cubes because

Rewrite the sum of cubes as

Analysis After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be factored further. However, the trinomial portion cannot be factored, so we do not need to check.

TRY IT #6 Factor the sum of cubes:

EXAMPLE 7

Factoring a Difference of Cubes Factor

Solution Notice that and

are cubes because

and

Write the difference of cubes as

Analysis Just as with the sum of cubes, we will not be able to further factor the trinomial portion.

TRY IT #7 Factor the difference of cubes:

Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest

power. These expressions follow the same factoring rules as those with integer exponents. For instance,

can

be factored by pulling out and being rewritten as

66 1 • Prerequisites
EXAMPLE 8 Factoring an Expression with Fractional or Negative Exponents Factor
Solution Factor out the term with the lowest value of the exponent. In this case, that would be

TRY IT #8 Factor
MEDIA Access these online resources for additional instruction and practice with factoring polynomials. Identify GCF (http://openstax.org/l/findgcftofact) Factor Trinomials when a Equals 1 (http://openstax.org/l/facttrinom1) Factor Trinomials when a is not equal to 1 (http://openstax.org/l/facttrinom2) Factor Sum or Difference of Cubes (http://openstax.org/l/sumdifcube)

1.5 SECTION EXERCISES
Verbal

1. If the terms of a polynomial do not have a GCF, does that mean it is not factorable? Explain.

2. A polynomial is factorable, but it is not a perfect square trinomial or a difference of two squares. Can you factor the polynomial without finding the GCF?

3. How do you factor by grouping?

Algebraic

For the following exercises, find the greatest common factor.

4.

5.

6.

7.

8.

9.

For the following exercises, factor by grouping.

10.

11.

12.

13.

14.

15.

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For the following exercises, factor the polynomial.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

For the following exercises, factor the polynomials.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

1.5 • Factoring Polynomials 67

Real-World Applications

For the following exercises, consider this scenario:

Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains

in one of the city’s parks. The park is a rectangle with an area of

m2, as shown in the figure below. The

length and width of the park are perfect factors of the area.

68 1 • Prerequisites

51. Factor by grouping to find the length and width of the park.

52. A statue is to be placed in the center of the park. The area of the base of the statue is Factor the area to find the lengths of the sides of the statue.

53. At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is Factor the area to find the lengths of the sides of the fountain.

For the following exercise, consider the following scenario:

A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure

below. The flagpole will take up a square plot with area

yd2.

54. Find the length of the base of the flagpole by factoring.

Extensions

For the following exercises, factor the polynomials completely.

55.

56.

57.

58.

59.

1.6 Rational Expressions

Learning Objectives
In this section, you will: Simplify rational expressions. Multiply rational expressions. Divide rational expressions. Add and subtract rational expressions. Simplify complex rational expressions.

A pastry shop has fixed costs of

per week and variable costs of per box of pastries. The shop’s costs per week in

terms of the number of boxes made, is

We can divide the costs per week by the number of boxes made to

determine the cost per box of pastries.

Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.
Simplifying Rational Expressions
The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to

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1.6 • Rational Expressions 69
rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.
We can factor the numerator and denominator to rewrite the expression.
Then we can simplify that expression by canceling the common factor
...
HOW TO Given a rational expression, simplify it. 1. Factor the numerator and denominator. 2. Cancel any common factors.
EXAMPLE 1 Simplifying Rational Expressions Simplify
Solution

Analysis We can cancel the common factor because any expression divided by itself is equal to 1.

Q&A

Can the term be cancelled in Example 1?
No. A factor is an expression that is multiplied by another expression. The numerator or the denominator.

term is not a factor of the

TRY IT #1 Simplify

Multiplying Rational Expressions
Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.
...
HOW TO
Given two rational expressions, multiply them.
1. Factor the numerator and denominator.

70 1 • Prerequisites
2. Multiply the numerators. 3. Multiply the denominators. 4. Simplify.
EXAMPLE 2 Multiplying Rational Expressions Multiply the rational expressions and show the product in simplest form:
Solution

TRY IT #2 Multiply the rational expressions and show the product in simplest form:

Dividing Rational Expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would

rewrite

as the product

Once the division expression has been rewritten as a multiplication expression,

we can multiply as we did before.

...
HOW TO
Given two rational expressions, divide them. 1. Rewrite as the first rational expression multiplied by the reciprocal of the second. 2. Factor the numerators and denominators. 3. Multiply the numerators. 4. Multiply the denominators. 5. Simplify.
EXAMPLE 3
Dividing Rational Expressions Divide the rational expressions and express the quotient in simplest form:

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Solution

1.6 • Rational Expressions 71

TRY IT #3 Divide the rational expressions and express the quotient in simplest form:
Adding and Subtracting Rational Expressions
Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.

We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.

The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest

multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions

and multiply all of the distinct factors. For instance, if the factored denominators were

and

then the LCD would be

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the

LCD. We would need to multiply the expression with a denominator of

by

and the expression with a

denominator of

by

...
HOW TO

Given two rational expressions, add or subtract them.
1. Factor the numerator and denominator. 2. Find the LCD of the expressions. 3. Multiply the expressions by a form of 1 that changes the denominators to the LCD. 4. Add or subtract the numerators. 5. Simplify.

EXAMPLE 4
Adding Rational Expressions Add the rational expressions:

Solution

72 1 • Prerequisites

First, we have to find the LCD. In this case, the LCD will be of 1 to obtain as the denominator for each fraction.

We then multiply each expression by the appropriate form

Now that the expressions have the same denominator, we simply add the numerators to find the sum.
Analysis Multiplying by or does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.
EXAMPLE 5 Subtracting Rational Expressions Subtract the rational expressions:
Solution

Q&A

Do we have to use the LCD to add or subtract rational expressions? No. Any common denominator will work, but it is easiest to use the LCD.

TRY IT #4 Subtract the rational expressions:

Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the

denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as

single rational expressions and dividing. The complex rational expression

can be simplified by rewriting the

numerator as the fraction and combining the expressions in the denominator as

We can then rewrite the

expression as a multiplication problem using the reciprocal of the denominator. We get

which is equal to

...
HOW TO

Given a complex rational expression, simplify it.

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1.6 • Rational Expressions 73

1. Combine the expressions in the numerator into a single rational expression by adding or subtracting. 2. Combine the expressions in the denominator into a single rational expression by adding or subtracting. 3. Rewrite as the numerator divided by the denominator. 4. Rewrite as multiplication. 5. Multiply. 6. Simplify.

EXAMPLE 6

Simplifying Complex Rational Expressions

Simplify:

.

Solution Begin by combining the expressions in the numerator into one expression.
  

  
Now the numerator is a single rational expression and the denominator is a single rational expression.

We can rewrite this as division, and then multiplication.

TRY IT #5 Simplify:

Q&A

Can a complex rational expression always be simplified? Yes. We can always rewrite a complex rational expression as a simplified rational expression.

MEDIA
Access these online resources for additional instruction and practice with rational expressions.
Simplify Rational Expressions (http://openstax.org/l/simpratexpress) Multiply and Divide Rational Expressions (http://openstax.org/l/multdivratex) Add and Subtract Rational Expressions (http://openstax.org/l/addsubratex) Simplify a Complex Fraction (http://openstax.org/l/complexfract)

74 1 • Prerequisites

1.6 SECTION EXERCISES
Verbal

1. How can you use factoring to simplify rational expressions?

2. How do you use the LCD to combine two rational expressions?

3. Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.

Algebraic

For the following exercises, simplify the rational expressions.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

For the following exercises, multiply the rational expressions and express the product in simplest form.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

For the following exercises, divide the rational expressions.

24.

25.

26.

27.

28.

29.

30.

31.

32.

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For the following exercises, add and subtract the rational expressions, and then simplify.

33.

34.

35.

36.

37.

38.

39.

40.

41.

For the following exercises, simplify the rational expression.

42.

43.

44.

45.

46.

47.

1.6 • Rational Expressions 75

48.

49.

50.

Real-World Applications

51. Brenda is placing tile on her

52. The area of Lijuan's yard is

bathroom floor. The area of the

ft2. A patch of

floor is

ft2. The area

sod has an area of

of one tile is

To find

ft2. Divide

the number of tiles needed,

the two areas and simplify

simplify the rational expression:

to find how many pieces of

sod Lijuan needs to cover

her yard.

53. Elroi wants to mulch his garden. His garden is ft2. One bag
of mulch covers ft2. Divide the expressions and simplify to find how many bags of mulch Elroi needs to mulch his garden.

Extensions
For the following exercises, perform the given operations and simplify.

54.

55.

56.

57.

76 1 • Chapter Review

Chapter Review

Key Terms

algebraic expression constants and variables combined using addition, subtraction, multiplication, and division

associative property of addition the sum of three numbers may be grouped differently without affecting the result;

in symbols,

associative property of multiplication the product of three numbers may be grouped differently without affecting

the result; in symbols,

base in exponential notation, the expression that is being multiplied

binomial a polynomial containing two terms

coefficient any real number in a polynomial in the form

commutative property of addition two numbers may be added in either order without affecting the result; in

symbols,

commutative property of multiplication two numbers may be multiplied in any order without affecting the result; in

symbols,

constant a quantity that does not change value

degree the highest power of the variable that occurs in a polynomial

difference of squares the binomial that results when a binomial is multiplied by a binomial with the same terms, but

the opposite sign

distributive property the product of a factor times a sum is the sum of the factor times each term in the sum; in

symbols,

equation a mathematical statement indicating that two expressions are equal

exponent in exponential notation, the raised number or variable that indicates how many times the base is being

multiplied

exponential notation a shorthand method of writing products of the same factor

factor by grouping a method for factoring a trinomial in the form

by dividing the x term into the sum of

two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire

expression

formula an equation expressing a relationship between constant and variable quantities

greatest common factor the largest polynomial that divides evenly into each polynomial

identity property of addition there is a unique number, called the additive identity, 0, which, when added to a

number, results in the original number; in symbols,

identity property of multiplication there is a unique number, called the multiplicative identity, 1, which, when

multiplied by a number, results in the original number; in symbols,

index the number above the radical sign indicating the nth root

integers the set consisting of the natural numbers, their opposites, and 0:

inverse property of addition for every real number there is a unique number, called the additive inverse (or

opposite), denoted which, when added to the original number, results in the additive identity, 0; in symbols,

inverse property of multiplication for every non-zero real number there is a unique number, called the multiplicative inverse (or reciprocal), denoted which, when multiplied by the original number, results in the
multiplicative identity, 1; in symbols,
irrational numbers the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers
leading coefficient the coefficient of the leading term leading term the term containing the highest degree least common denominator the smallest multiple that two denominators have in common monomial a polynomial containing one term natural numbers the set of counting numbers: order of operations a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to
operations perfect square trinomial the trinomial that results when a binomial is squared polynomial a sum of terms each consisting of a variable raised to a nonnegative integer power principal nth root the number with the same sign as that when raised to the nth power equals principal square root the nonnegative square root of a number that, when multiplied by itself, equals radical the symbol used to indicate a root radical expression an expression containing a radical symbol radicand the number under the radical symbol

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1 • Chapter Review 77

rational expression the quotient of two polynomial expressions

rational numbers the set of all numbers of the form where and are integers and

Any rational number

may be written as a fraction or a terminating or repeating decimal.

real number line a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent

0; positive numbers lie to the right of 0 and negative numbers to the left.

real numbers the sets of rational numbers and irrational numbers taken together

scientific notation a shorthand notation for writing very large or very small numbers in the form

where

and is an integer

term of a polynomial any

of a polynomial in the form

trinomial a polynomial containing three terms

variable a quantity that may change value

whole numbers the set consisting of 0 plus the natural numbers:

Key Equations

Rules of Exponents For nonzero real numbers and and integers and

Product rule

Quotient rule

Power rule Zero exponent rule

Negative rule

Power of a product rule

Power of a quotient rule

perfect square trinomial
difference of squares
difference of squares
perfect square trinomial
sum of cubes
difference of cubes
Key Concepts
1.1 Real Numbers: Algebra Essentials
• Rational numbers may be written as fractions or terminating or repeating decimals. See Example 1 and Example 2. • Determine whether a number is rational or irrational by writing it as a decimal. See Example 3. • The rational numbers and irrational numbers make up the set of real numbers. See Example 4. A number can be
classified as natural, whole, integer, rational, or irrational. See Example 5.

78 1 • Chapter Review
• The order of operations is used to evaluate expressions. See Example 6. • The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of
real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See Example 7. • Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See Example 8. They take on a numerical value when evaluated by replacing variables with constants. See Example 9, Example 10, and Example 12 • Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See Example 11 and Example 13.
1.2 Exponents and Scientific Notation
• Products of exponential expressions with the same base can be simplified by adding exponents. See Example 1. • Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See Example
2. • Powers of exponential expressions with the same base can be simplified by multiplying exponents. See Example 3. • An expression with exponent zero is defined as 1. See Example 4. • An expression with a negative exponent is defined as a reciprocal. See Example 5 and Example 6. • The power of a product of factors is the same as the product of the powers of the same factors. See Example 7. • The power of a quotient of factors is the same as the quotient of the powers of the same factors. See Example 8. • The rules for exponential expressions can be combined to simplify more complicated expressions. See Example 9. • Scientific notation uses powers of 10 to simplify very large or very small numbers. See Example 10 and Example 11. • Scientific notation may be used to simplify calculations with very large or very small numbers. See Example 12 and
Example 13.
1.3 Radicals and Rational Exponents
• The principal square root of a number is the nonnegative number that when multiplied by itself equals See Example 1.
• If and are nonnegative, the square root of the product is equal to the product of the square roots of and See Example 2 and Example 3.
• If and are nonnegative, the square root of the quotient is equal to the quotient of the square roots of and See Example 4 and Example 5.
• We can add and subtract radical expressions if they have the same radicand and the same index. See Example 6 and Example 7.
• Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See Example 8 and Example 9.
• The principal nth root of is the number with the same sign as that when raised to the nth power equals These roots have the same properties as square roots. See Example 10.
• Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See Example 11 and Example 12.
• The properties of exponents apply to rational exponents. See Example 13.
1.4 Polynomials
• A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See Example 1.
• We can add and subtract polynomials by combining like terms. See Example 2 and Example 3. • To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in
the second. Then add the products. See Example 4. • FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See Example 5. • Perfect square trinomials and difference of squares are special products. See Example 6 and Example 7. • Follow the same rules to work with polynomials containing several variables. See Example 8.
1.5 Factoring Polynomials
• The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See Example 1.
• Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See Example 2.
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1 • Exercises 79

• Trinomials can be factored using a process called factoring by grouping. See Example 3. • Perfect square trinomials and the difference of squares are special products and can be factored using equations.
See Example 4 and Example 5. • The sum of cubes and the difference of cubes can be factored using equations. See Example 6 and Example 7. • Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See Example 8.

1.6 Rational Expressions

• Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See Example 1.
• We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See Example 2.
• To divide rational expressions, multiply by the reciprocal of the second expression. See Example 3. • Adding or subtracting rational expressions requires finding a common denominator. See Example 4 and Example 5. • Complex rational expressions have fractions in the numerator or the denominator. These expressions can be
simplified. See Example 6.
Exercises
Review Exercises

Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.

1.

2.

3.

For the following exercises, solve the equation.

4.

5.

For the following exercises, simplify the expression.

6.

7.

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

8. 11

9. 0

10.

11.

Exponents and Scientific Notation For the following exercises, simplify the expression.

12.

13.

15.

16.

18.

19.

14.
17.
20. Write the number in standard notation:

80 1 • Exercises
21. Write the number in scientific notation: 16,340,000

Radicals and Rational Expressions

For the following exercises, find the principal square root.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

Polynomials

For the following exercises, perform the given operations and simplify.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

Factoring Polynomials

For the following exercises, find the greatest common factor.

45.

46.

47.

For the following exercises, factor the polynomial.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

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60.

Rational Expressions For the following exercises, simplify the expression.

61.

62.

63.

64.

65.

66.

67.

68.

69.

1 • Exercises 81

70.

Practice Test
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

1.

2.

For the following exercises, evaluate the expression.

3.

4.

5. Write the number in standard notation:

6. Write the number in scientific notation: 0.0000000212.

For the following exercises, simplify the expression.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

82 1 • Exercises

21.

22.

For the following exercises, factor the polynomial.

23.

24.

25.

26.

For the following exercises, simplify the expression.

27.

28.

29.

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2 • Introduction 83

2

EQUATIONS AND INEQUALITIES

From the air, a landscape of circular crop fields may seem random, but they are laid out and irrigated very precisely. Farmers and irrigation providers combining agricultural science, engineering, and mathematics to achieve the most productive and efficient array. (Credit: Modification of "Aerial Phot of Center Pivot Irrigations Systems (1)" by Soil Science/flickr)
Chapter Outline
2.1 The Rectangular Coordinate Systems and Graphs 2.2 Linear Equations in One Variable 2.3 Models and Applications 2.4 Complex Numbers 2.5 Quadratic Equations 2.6 Other Types of Equations 2.7 Linear Inequalities and Absolute Value Inequalities
Introduction to Equations and Inequalities
Irrigation is a critical aspect of agriculture, which can expand the yield of farms and enable farming in areas not naturally viable for crops. But the materials, equipment, and the water itself are expensive and complex. To be efficient and productive, farm owners and irrigation specialists must carefully lay out the network of pipes, pumps, and related equipment. The available land can be divided into regular portions (similar to a grid), and the different sizes of irrigation systems and conduits can be installed within the plotted area.
2.1 The Rectangular Coordinate Systems and Graphs
Learning Objectives
In this section, you will: Plot ordered pairs in a Cartesian coordinate system. Graph equations by plotting points. Graph equations with a graphing utility. Find x-intercepts and y-intercepts. Use the distance formula. Use the midpoint formula.

84 2 • Equations and Inequalities

Figure 1
Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in Figure 1. Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.
Plotting Ordered Pairs in the Cartesian Coordinate System
An old story describes how seventeenth-century philosopher/mathematician René Descartes, while sick in bed, invented the system that has become the foundation of algebra. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly’s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers—the displacement from the horizontal axis and the displacement from the vertical axis.
While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, it was Descartes who introduced the components that comprise the Cartesian coordinate system, a grid system having perpendicular axes. Descartes named the horizontal axis the x-axis and the vertical axis the y-axis.
The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the x-axis and the y-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant; the quadrants are numbered counterclockwise as shown in Figure 2

Figure 2

The center of the plane is the point at which the two axes cross. It is known as the origin, or point

From the origin,

each axis is further divided into equal units: increasing, positive numbers to the right on the x-axis and up the y-axis;

decreasing, negative numbers to the left on the x-axis and down the y-axis. The axes extend to positive and negative

infinity as shown by the arrowheads in Figure 3.

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2.1 • The Rectangular Coordinate Systems and Graphs 85

Figure 3

Each point in the plane is identified by its x-coordinate, or horizontal displacement from the origin, and its

y-coordinate, or vertical displacement from the origin. Together, we write them as an ordered pair indicating the

combined distance from the origin in the form

An ordered pair is also known as a coordinate pair because it

consists of x- and y-coordinates. For example, we can represent the point

in the plane by moving three units to

the right of the origin in the horizontal direction, and one unit down in the vertical direction. See Figure 4.

Figure 4
When dividing the axes into equally spaced increments, note that the x-axis may be considered separately from the y-axis. In other words, while the x-axis may be divided and labeled according to consecutive integers, the y-axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.

Cartesian Coordinate System

A two-dimensional plane where the

• x-axis is the horizontal axis • y-axis is the vertical axis

A point in the plane is defined as an ordered pair,

such that x is determined by its horizontal distance from the

origin and y is determined by its vertical distance from the origin.

EXAMPLE 1

Plotting Points in a Rectangular Coordinate System

Plot the points

and

in the plane.

Solution

To plot the point

begin at the origin. The x-coordinate is –2, so move two units to the left. The y-coordinate is 4,

so then move four units up in the positive y direction.

86 2 • Equations and Inequalities

To plot the point

begin again at the origin. The x-coordinate is 3, so move three units to the right. The

y-coordinate is also 3, so move three units up in the positive y direction.

To plot the point

begin again at the origin. The x-coordinate is 0. This tells us not to move in either direction

along the x-axis. The y-coordinate is –3, so move three units down in the negative y direction. See the graph in Figure 5.

Figure 5
Analysis Note that when either coordinate is zero, the point must be on an axis. If the x-coordinate is zero, the point is on the y-axis. If the y-coordinate is zero, the point is on the x-axis.

Graphing Equations by Plotting Points

We can plot a set of points to represent an equation. When such an equation contains both an x variable and a y variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any graph on a twodimensional plane is a graph in two variables.

Suppose we want to graph the equation

We can begin by substituting a value for x into the equation and

determining the resulting value of y. Each pair of x- and y-values is an ordered pair that can be plotted. Table 1 lists

values of x from –3 to 3 and the resulting values for y.

Table 1 We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See Figure 6. This is not true for all equations.
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2.1 • The Rectangular Coordinate Systems and Graphs 87

Figure 6
Note that the x-values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of x to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.
...
HOW TO
Given an equation, graph by plotting points.
1. Make a table with one column labeled x, a second column labeled with the equation, and a third column listing the resulting ordered pairs.
2. Enter x-values down the first column using positive and negative values. Selecting the x-values in numerical order will make the graphing simpler.
3. Select x-values that will yield y-values with little effort, preferably ones that can be calculated mentally. 4. Plot the ordered pairs. 5. Connect the points if they form a line.

EXAMPLE 2

Graphing an Equation in Two Variables by Plotting Points

Graph the equation

by plotting points.

Solution First, we construct a table similar to Table 2. Choose x values and calculate y.

Table 2

88 2 • Equations and Inequalities
Table 2 Now, plot the points. Connect them if they form a line. See Figure 7

Figure 7

TRY IT #1 Construct a table and graph the equation by plotting points:

Graphing Equations with a Graphing Utility

Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be

manipulated so they are written in the style

The TI-84 Plus, and many other calculator makes and models,

have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of

a graph can be seen.

For example, the equation

has been entered in the TI-84 Plus shown in Figure 8a. In Figure 8b, the resulting

graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen

on the TI-84 Plus shows

and

See Figure 8c.

Figure 8 a. Enter the equation. b. This is the graph in the original window. c. These are the original settings.
By changing the window to show more of the positive x-axis and more of the negative y-axis, we have a much better view of the graph and the x- and y-intercepts. See Figure 9a and Figure 9b.

Figure 9 a. This screen shows the new window settings. b. We can clearly view the intercepts in the new window.

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2.1 • The Rectangular Coordinate Systems and Graphs 89
EXAMPLE 3 Using a Graphing Utility to Graph an Equation Use a graphing utility to graph the equation:
Solution Enter the equation in the y= function of the calculator. Set the window settings so that both the x- and y- intercepts are showing in the window. See Figure 10.
Figure 10
Finding x-intercepts and y-intercepts
The intercepts of a graph are points at which the graph crosses the axes. The x-intercept is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The y-intercept is the point at which the graph crosses the y-axis. At this point, the x-coordinate is zero. To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y. For example, lets find the intercepts of the equation To find the x-intercept, set
To find the y-intercept, set
We can confirm that our results make sense by observing a graph of the equation as in Figure 11. Notice that the graph crosses the axes where we predicted it would.

Figure 11

Given an equation, find the intercepts.

• Find the x-intercept by setting

and solving for

90 2 • Equations and Inequalities

• Find the y-intercept by setting

and solving for

EXAMPLE 4

Finding the Intercepts of the Given Equation

Find the intercepts of the equation

Then sketch the graph using only the intercepts.

Solution

Set

to find the x-intercept.

Set

to find the y-intercept.

Plot both points, and draw a line passing through them as in Figure 12.

Figure 12

TRY IT #2 Find the intercepts of the equation and sketch the graph:

Using the Distance Formula

Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the

plane. The Pythagorean Theorem,

is based on a right triangle where a and b are the lengths of the legs

adjacent to the right angle, and c is the length of the hypotenuse. See Figure 13.

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Figure 13