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College Algebra Fifth Edition Robert E. Moyer, PhD Associate Professor of Mathematics Southwest Minnesota State University (Retired) Professor and Chairman, Department of Mathematics & Physics Fort Valley State University (Retired) Murray R. Spiegel, PhD Former Professor and Chairman, Mathematics Department Rensselaer Polytechnic Institute, Hartford Graduate Center Schaum’s Outline Series New York Chicago San Francisco Athens London Madrid Mexico City Milan New Delhi Singapore Sydney Toronto 00_Speigel_FM_i-xii.indd 1 17/08/18 2:55 PM DR. ROBERT E. MOYER taught mathematics and mathematics education at Southwest Minnesota State University in Marshall, Minnesota, from 2002 to 2009 and served as an adjunct professor of mathematics there from 2009 to 2012. Before coming to SMSU, he taught mathematics and mathematics education at Fort Valley State University in Fort Valley, Georgia, from 1985 to 2000. He served as head of the Department of Mathematics and Physics from 1992 to 1994. Prior to teaching at the university level, Dr. Moyer served as the K-12 mathematics consultant for seven years at Middle Geor- gia Regional Educational Service Agency, a five-county education cooperative in central Georgia. Dr. Moyer taught high school mathematics for seven years in Rantoul, Illinois, and for five years in Carmi, Illinois. He has developed and taught numerous in- service courses for mathematics teachers. He received his Doctor of Philosophy in Mathematics Education from the University of Illinois (Urbana-Champaign) in 1974. He received his Master of Science in 1967 and his Bachelor of Science in 1964, both in Mathematics Education from Southern Illinois University (Carbondale). MURRAY R. SPIEGEL received the MS degree in Physics and the PhD in Mathematics from Cornell University. He had posi- tions at Harvard University, Columbia University, Oak Ridge, and Rensselaer Polytechnic Institute, and had served as mathemati- cal consultant at several large companies. His last position was Professor and Chairman of Mathematics at the Rensselaer Poly- technic Institute, Hartford Graduate Center. He was interested in most branches of mathematics, especially those which involved applications to physics and engineering problems. He was the author of numerous journal articles and 14 books on various topics in mathematics. Copyright © 2019 by McGraw-Hill Education. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-1-26-012077-6 MHID: 1-26-012077-5 The material in this eBook also appears in the print version of this title: ISBN: 978-1-26-012076-9, MHID: 1-26-012076-7. eBook conversion by codeMantra Version 1.0 All trademarks are trademarks of their respective owners. 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Under no circumstances shall McGraw-Hill Education and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. Preface In the fifth edition, the comprehensiveness of earlier editions is retained and three new chapters are added so that all of the topics commonly taught in college algebra are contained in a single source. Although the use of logarithm tables and determinants has declined, these topics are retained for use by instructors who still include them in their courses. Manual solutions are used to develop an understanding of the procedures that are fre- quently done using calculators and computers. The book is complete in itself and can be used equally well by those who are in a class studying college algebra for the first time as well as those who wish to review the fundamental principles and procedures of college algebra on their own. Students who are studying advanced algebra in high school will be able to use the book as a source of additional examples, explanations, and problems. The thorough treatment of the topics of algebra allows an instructor to use the book as the textbook for a course, as a resource for material on a specific topic, or as a source for additional problems. Each of the chapters 1 through 33 contains a summary of the necessary definitions and theorems followed by a set of solved problems and concluding with a set of supplementary problems with answers. Chapter 32 introduces three additional procedures for approximating the real zeros of polynomial equations of degree three or more. Chapter 33 is an informal development of the basic calculus concepts of limit, continuity, and conver- gence using the algebra procedures from the earlier chapters. Chapter 34 contains additional solved problems and supplementary problems with answers for each of the prior chapters. The choice of whether to use a calculator or not is left to the student. A calculator is not required, but it can be used in conjunction with the book. There are no directions on how to use a graphing calculator to do the problems, but there are several instances of the general procedures to be used and the student needs to consult the manual for the calculator being used to see how to implement the procedures on that particular calculator. Dr. Robert E. Moyer Associate Professor of Mathematics Southwest Minnesota State University (Retired) Professor of Mathematics Fort Valley State University (Retired) iii 00_Speigel_FM_i-xii.indd 3 17/08/18 2:55 PM This page intentionally left blank 00_Speigel_FM_i-xii.indd 4 17/08/18 2:55 PM Contents CHAPTER 1 Fundamental Operations with Numbers 1 1.1 Four Operations 1 1.2 System of Real Numbers 2 1.3 Graphical Representation of Real Numbers 2 1.4 Properties of Addition and Multiplication of Real Numbers 3 1.5 Rules of Signs 3 1.6 Exponents and Powers 4 1.7 Operations with Fractions 4 Solved Problems 5 Supplementary Problems 9 CHAPTER 2 Fundamental Operations with Algebraic Expressions 12 2.1 Algebraic Expressions 12 2.2 Terms 12 2.3 Degree 13 2.4 Grouping 13 2.5 Computation with Algebraic Expressions 13 Solved Problems 16 Supplementary Problems 20 CHAPTER 3 Properties of Numbers 22 3.1 Sets of Numbers 22 3.2 Properties 22 3.3 Additional Properties 23 Solved Problems 23 Supplementary Problems 25 CHAPTER 4 Special Products 27 4.1 Special Products 27 4.2 Products Yielding Answers of the Form an ± bn 28 Solved Problems 28 Supplementary Problems 30 CHAPTER 5 Factoring 32 5.1 Factoring 32 5.2 Factorization Procedures 33 5.3 Greatest Common Factor 34 5.4 Least Common Multiple 34 Solved Problems 35 Supplementary Problems 39 v 00_Speigel_FM_i-xii.indd 5 17/08/18 2:55 PM vi Contents CHAPTER 6 Fractions 41 6.1 Rational Algebraic Fractions 41 6.2 Operations with Algebraic Fractions 42 6.3 Complex Fractions 43 Solved Problems 44 Supplementary Problems 46 CHAPTER 7 Exponents 48 7.1 Positive Integral Exponent 48 7.2 Negative Integral Exponent 48 7.3 Roots 48 7.4 Rational Exponents 49 7.5 General Laws of Exponents 49 7.6 Scientific Notation 50 Solved Problems 50 Supplementary Problems 56 CHAPTER 8 Radicals 58 8.1 Radical Expressions 58 8.2 Laws for Radicals 58 8.3 Simplifying Radicals 58 8.4 Operations with Radicals 59 8.5 Rationalizing Binomial Denominators 60 Solved Problems 61 Supplementary Problems 65 CHAPTER 9 Simple Operations with Complex Numbers 67 9.1 Complex Numbers 67 9.2 Graphical Representation of Complex Numbers 67 9.3 Algebraic Operations with Complex Numbers 68 Solved Problems 69 Supplementary Problems 71 CHAPTER 10 Equations in General 73 10.1 Equations 73 10.2 Operations Used in Transforming Equations 73 10.3 Equivalent Equations 74 10.4 Formulas 74 10.5 Polynomial Equations 75 Solved Problems 75 Supplementary Problems 79 CHAPTER 11 Ratio, Proportion, and Variation 81 11.1 Ratio 81 11.2 Proportion 81 11.3 Variation 81 11.4 Unit Price 82 00_Speigel_FM_i-xii.indd 6 17/08/18 2:55 PM Contents vii 11.5 Best Buy 82 Solved Problems 83 Supplementary Problems 86 CHAPTER 12 Functions and Graphs 89 12.1 Variables 89 12.2 Relations 89 12.3 Functions 89 12.4 Function Notation 90 12.5 Rectangular Coordinate System 90 12.6 Function of Two Variables 91 12.7 Symmetry 91 12.8 Shifts 92 12.9 Scaling 93 12.10 Using a Graphing Calculator 93 Solved Problems 96 Supplementary Problems 106 CHAPTER 13 Linear Equations in One Variable 114 13.1 Linear Equations 114 13.2 Literal Equations 114 13.3 Word Problems 115 Solved Problems 116 Supplementary Problems 124 CHAPTER 14 Equations of Lines 128 14.1 Slope of a Line 128 14.2 Parallel and Perpendicular Lines 129 14.3 Slope-Intercept Form of Equation of a Line 130 14.4 Slope-Point Form of Equation of a Line 130 14.5 Two-Point Form of Equation of a Line 130 14.6 Intercept Form of Equation of a Line 131 Solved Problems 131 Supplementary Problems 134 CHAPTER 15 Simultaneous Linear Equations 137 15.1 Systems of Two Linear Equations 137 15.2 Systems of Three Linear Equations 138 Solved Problems 139 Supplementary Problems 146 CHAPTER 16 Quadratic Equations in One Variable 150 16.1 Quadratic Equations 150 16.2 Methods of Solving Quadratic Equations 150 16.3 Sum and Product of the Roots 152 16.4 Nature of the Roots 152 16.5 Radical Equations 152 16.6 Quadratic-Type Equations 153 00_Speigel_FM_i-xii.indd 7 17/08/18 2:55 PM viii Contents Solved Problems 153 Supplementary Problems 163 CHAPTER 17 Conic Sections 169 17.1 General Quadratic Equations 169 17.2 Conic Sections 170 17.3 Circles 170 17.4 Parabolas 171 17.5 Ellipses 173 17.6 Hyperbolas 177 17.7 Graphing Conic Sections with a Calculator 180 Solved Problems 180 Supplementary Problems 186 CHAPTER 18 Systems of Equations Involving Quadratics 191 18.1 Graphical Solution 191 18.2 Algebraic Solution 191 Solved Problems 193 Supplementary Problems 197 CHAPTER 19 Inequalities 199 19.1 Definitions 199 19.2 Principles of Inequalities 199 19.3 Absolute Value Inequalities 200 19.4 Higher Degree Inequalities 200 19.5 Linear Inequalities in Two Variables 202 19.6 Systems of Linear Inequalities 202 19.7 Linear Programming 203 Solved Problems 204 Supplementary Problems 210 CHAPTER 20 Polynomial Functions 214 20.1 Polynomial Equations 214 20.2 Zeros of Polynomial Equations 214 20.3 Solving Polynomial Equations 216 20.4 Approximating Real Zeros 218 Solved Problems 219 Supplementary Problems 231 CHAPTER 21 Rational Functions 235 21.1 Rational Functions 235 21.2 Vertical Asymptotes 235 21.3 Horizontal Asymptotes 235 21.4 Graphing Rational Functions 236 21.5 Graphing Rational Functions Using a Graphing Calculator 238 Solved Problems 238 Supplementary Problems 240 00_Speigel_FM_i-xii.indd 8 17/08/18 2:55 PM Contents ix CHAPTER 22 Sequences and Series 245 22.1 Sequences 245 22.2 Arithmetic Sequences 245 22.3 Geometric Sequences 245 22.4 Infinite Geometric Series 246 22.5 Harmonic Sequences 246 22.6 Means 246 Solved Problems 247 Supplementary Problems 258 CHAPTER 23 Logarithms 263 23.1 Definition of a Logarithm 263 23.2 Laws of Logarithms 263 23.3 Common Logarithms 264 23.4 Using a Common Logarithm Table 264 23.5 Natural Logarithms 265 23.6 Using a Natural Logarithms Table 265 23.7 Finding Logarithms Using a Calculator 266 Solved Problems 267 Supplementary Problems 273 CHAPTER 24 Applications of Logarithms and Exponents 276 24.1 Introduction 276 24.2 Simple Interest 276 24.3 Compound Interest 277 24.4 Applications of Logarithms 278 24.5 Applications of Exponents 280 Solved Problems 280 Supplementary Problems 284 CHAPTER 25 Permutations and Combinations 288 25.1 Fundamental Counting Principle 288 25.2 Permutations 288 25.3 Combinations 289 25.4 Using a Calculator 290 Solved Problems 290 Supplementary Problems 300 CHAPTER 26 The Binomial Theorem 303 26.1 Combinatorial Notation 303 26.2 Expansion of (a + x)n 303 Solved Problems 304 Supplementary Problems 308 CHAPTER 27 Probability 310 27.1 Simple Probability 310 27.2 Compound Probability 310 00_Speigel_FM_i-xii.indd 9 17/08/18 2:55 PM