advanced algebra

Advanced Algebra v 1.0 This is the book Advanced Algebra (v 1.0) This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/ 3.0/) license See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and make it available to everyone else under the same terms This book was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz (http://lardbucket.org) in an effort to preserve the availability of this book Normally, the author and publisher would be credited here However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed Additionally, per the publisher's request, their name has been removed in some passages More information is available on this project's attribution page (http://2012books.lardbucket.org/attribution.html?utm_source=header) For more information on the source of this book, or why it is available for free, please see the project's home page (http://2012books.lardbucket.org/) You can browse or download additional books there ii Table of Contents About the Author Acknowledgments Preface Chapter 1: Algebra Fundamentals Review of Real Numbers and Absolute Value Operations with Real Numbers 35 Square and Cube Roots of Real Numbers 68 Algebraic Expressions and Formulas 95 Rules of Exponents and Scientific Notation 125 Polynomials and Their Operations 158 Solving Linear Equations 195 Solving Linear Inequalities with One Variable 232 Review Exercises and Sample Exam 261 Chapter 2: Graphing Functions and Inequalities 281 Relations, Graphs, and Functions 282 Linear Functions and Their Graphs 325 Modeling Linear Functions 367 Graphing the Basic Functions 400 Using Transformations to Graph Functions 434 Solving Absolute Value Equations and Inequalities 475 Solving Inequalities with Two Variables 512 Review Exercises and Sample Exam 545 Chapter 3: Solving Linear Systems 580 Linear Systems with Two Variables and Their Solutions 581 Solving Linear Systems with Two Variables 610 Applications of Linear Systems with Two Variables 647 Solving Linear Systems with Three Variables 675 Matrices and Gaussian Elimination 704 Determinants and Cramer’s Rule 735 Solving Systems of Inequalities with Two Variables 766 Review Exercises and Sample Exam 796 iii Chapter 4: Polynomial and Rational Functions 818 Algebra of Functions 819 Factoring Polynomials 853 Factoring Trinomials 889 Solve Polynomial Equations by Factoring 923 Rational Functions: Multiplication and Division 956 Rational Functions: Addition and Subtraction 986 Solving Rational Equations 1018 Applications and Variation 1051 Review Exercises and Sample Exam 1088 Chapter 5: Radical Functions and Equations 1111 Roots and Radicals 1112 Simplifying Radical Expressions 1149 Adding and Subtracting Radical Expressions 1179 Multiplying and Dividing Radical Expressions 1202 Rational Exponents 1239 Solving Radical Equations 1266 Complex Numbers and Their Operations 1302 Review Exercises and Sample Exam 1332 Chapter 6: Solving Equations and Inequalities 1353 Extracting Square Roots and Completing the Square 1354 Quadratic Formula 1390 Solving Equations Quadratic in Form 1421 Quadratic Functions and Their Graphs 1450 Solving Quadratic Inequalities 1493 Solving Polynomial and Rational Inequalities 1524 Review Exercises and Sample Exam 1548 Chapter 7: Exponential and Logarithmic Functions 1568 Composition and Inverse Functions 1569 Exponential Functions and Their Graphs 1605 Logarithmic Functions and Their Graphs 1639 Properties of the Logarithm 1675 Solving Exponential and Logarithmic Equations 1701 Applications 1735 Review Exercises and Sample Exam 1765 iv Chapter 8: Conic Sections 1788 Distance, Midpoint, and the Parabola 1789 Circles 1825 Ellipses 1854 Hyperbolas 1885 Solving Nonlinear Systems 1923 Review Exercises and Sample Exam 1946 Chapter 9: Sequences, Series, and the Binomial Theorem 1977 Introduction to Sequences and Series 1978 Arithmetic Sequences and Series 1999 Geometric Sequences and Series 2025 Binomial Theorem 2057 Review Exercises and Sample Exam 2074 v About the Author John Redden John Redden earned his degrees at California State University–Northridge and Glendale Community College He is now a professor of mathematics at the College of the Sequoias, located in Visalia, California With over a decade of experience working with students to develop their algebra skills, he knows just where they struggle and how to present complex techniques in more understandable ways His student-friendly and commonsense approach carries over to his writing of Intermediate Algebra and various other open-source learning resources Author site: http://edunettech.blogspot.com/ Acknowledgments I would like to thank the following reviewers whose feedback helped improve the final product: • • • • • • • • • • • • • • • • • • • Katherine Adams, Eastern Michigan University Sheri Berger, Los Angeles Valley College Seung Choi, Northern Virginia Community College Stephen DeLong, Colorado Mountain College Keith Eddy, College of the Sequoias Solomon Emeghara, William Patterson University Audrey Gillant, SUNY–Maritime Barbara Goldner, North Seattle Community College Joseph Grich, William Patterson University Caroll Hobbs, Pensacola State College Clark Ingham, Mott Community College Valerie LaVoice, NHTI, Concord Community College Sandra Martin, Brevard Schools Bethany Mueller, Pensacola State College Tracy Redden, College of the Sequoias James Riley, Northern Arizona University Bamdad Samii, California State University–Northridge Michael Scott, California State University–Monterey Bay Nora Wheeler, Santa Rosa Junior College I would also like to acknowledge Michael Boezi and Vanessa Gennarelli of Unnamed Publisher The success of this project is in large part due to their vision and expertise Finally, a special heartfelt thank-you is due to my wife, Tracy, who spent countless hours proofreading and editing these pages—all this while maintaining a tight schedule for our family Without her, this textbook would not have been possible Preface Intermediate Algebra is the second part of a two-part course in Algebra Written in a clear and concise manner, it carefully builds on the basics learned in Elementary Algebra and introduces the more advanced topics required for further study of applications found in most disciplines Used as a standalone textbook, it offers plenty of review as well as something new to engage the student in each chapter Written as a blend of the traditional and graphical approaches to the subject, this textbook introduces functions early and stresses the geometry behind the algebra While CAS independent, a standard scientific calculator will be required and further research using technology is encouraged Intermediate Algebra clearly lays out the steps required to build the skills needed to solve a variety of equations and interpret the results With robust and diverse exercise sets, students have the opportunity to solve plenty of practice problems In addition to embedded video examples and other online learning resources, the importance of practice with pencil and paper is stressed This text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today In addition, Intermediate Algebra was written from the ground up in an open and modular format, allowing the instructor to modify it and leverage their individual expertise as a means to maximize the student experience and success The importance of Algebra cannot be overstated; it is the basis for all mathematical modeling used in all disciplines After completing a course sequence based on Elementary and Intermediate Algebra, students will be on firm footing for success in higher-level studies at the college level Chapter Algebra Fundamentals Chapter Algebra Fundamentals 1.1 Review of Real Numbers and Absolute Value LEARNING OBJECTIVES Review the set of real numbers Review the real number line and notation Define the geometric and algebraic definition of absolute value Real Numbers Algebra is often described as the generalization of arithmetic The systematic use of variables1, letters used to represent numbers, allows us to communicate and solve a wide variety of real-world problems For this reason, we begin by reviewing real numbers and their operations A set2 is a collection of objects, typically grouped within braces { }, where each object is called an element3 When studying mathematics, we focus on special sets of numbers ℕ = {1, 2, 3, 4, 5, …} W = {0, 1, 2, 3, 4, 5, …} Natural Numbers Whole Numbers ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …}Integers Letters used to represent numbers Any collection of objects An object within a set A set consisting of elements that belong to a given set The three periods (…) are called an ellipsis and indicate that the numbers continue without bound A subset4, denoted ⊆, is a set consisting of elements that belong to a given set Notice that the sets of natural5 and whole numbers6 are both subsets of the set of integers and we can write: The set of counting numbers: {1, 2, 3, 4, 5, …} The set of natural numbers combined with zero: {0, 1, 2, 3, 4, 5, …} A subset with no elements, denoted Ø or { } ℕ ⊆ ℤ and W ⊆ ℤ A set with no elements is called the empty set7 and has its own special notation: Chapter Sequences, Series, and the Binomial Theorem REVIEW EXERCISES INTRODUCTION TO SEQUENCES AND SERIES Find the first terms of the sequence as well as the 30th term an = 5n − an = −4n + 3 an = −10n an = 3n an = (−1) n (n − 2)2 an = (−1)n 2n−1 an = 2n+1 n an = (−1) n+1 (n − 1) Find the first terms of the sequence an = nx n 2n+1 10 an = (−1)n−1 x n+2 n 11 an = n x 2n 12 an = (−3x) n−1 13 an = an−1 + where a1 = 14 an = 4an−1 + where a1 = −2 15 an = an−2 − 3an−1 where a1 = and a2 = −3 16 an = 5an−2 − an−1 where a1 = −1 and a2 = Find the indicated partial sum 17 1, 4, 7, 10, 13,…; S 18 3, 1, −1, −3, −5,…; S 9.5 Review Exercises and Sample Exam 2075 Chapter Sequences, Series, and the Binomial Theorem 19 −1, 3, −5, 7, −9,…; S 20 an = (−1) n n 2; S 21 an = −3(n − 2)2 ; S 22 an = (− 15 ) n−2 ; S4 Evaluate ∑( 23 k=1 24 ∑ k=1 25 ∑ n=1 27 5(−1) n−1 − k) ∑( k=4 28 (−1) k 3k n+1 ∑ n n=1 26 − 2k) 2 ∑ (3) k=−2 k ARITHMETIC SEQUENCES AND SERIES Write the first terms of the arithmetic sequence given its first term and common difference Find a formula for its general term 29 a1 = 6; d = 30 a1 = 5; d = 31 a1 = 5; d = −3 32 a1 = − ;d =− 33 a1 = − ;d =− 9.5 Review Exercises and Sample Exam 2076 Chapter Sequences, Series, and the Binomial Theorem 34 a1 = −3.6 ; d = 1.2 35 a1 = 7; d = 36 a1 = 1; d = Given the terms of an arithmetic sequence, find a formula for the general term 37 10, 20, 30, 40, 50,… 38 −7, −5, −3, −1, 1,… 39 −2, −5, −8, −11, −14,… 1 , 0, , , 1,… 3 40 − 41 a4 = 11 and a9 = 26 42 a5 = −5 and a10 = −15 43 a6 = and a24 = 15 44 a3 = −1.4 and a7 = Calculate the indicated sum given the formula for the general term of an arithmetic sequence 45 an = 4n − 3; S 60 46 an = −2n + 9; S 35 47 an = 48 an = −n + 49 an = 1.8n − 4.2 ; S 45 50 an = −6.5n + ; S 35 n− ;S 15 ;S 20 Evaluate ∑( 22 51 n=1 100 52 ∑ n=1 9.5 Review Exercises and Sample Exam 7n − 5) (1 − 4n) 2077 Chapter Sequences, Series, and the Binomial Theorem n ∑ (3 ) n=1 35 53 − n+1 ∑( ) n=1 30 54 ∑ 40 55 n=1 (2.3n − 1.1) 56 ∑ 300 n n=1 57 Find the sum of the first 175 positive odd integers 58 Find the sum of the first 175 positive even integers and a5 =− 59 Find all arithmetic means between a1 = 60 Find all arithmetic means between a3 = −7 and a7 = 13 61 A 5-year salary contract offers $58,200 for the first year with a $4,200 increase each additional year Determine the total salary obligation over the 5-year period 62 The first row of seating in a theater consists of 10 seats Each successive row consists of four more seats than the previous row If there are 14 rows, how many total seats are there in the theater? GEOMETRIC SEQUENCES AND SERIES Write the first terms of the geometric sequence given its first term and common ratio Find a formula for its general term 63 a1 = 5; r = 64 a1 = 3; r = −2 65 a1 = 1; r = − 66 a1 = −4; r = 67 a1 = 1.2; r = 0.2 68 a1 = −5.4 ; r = −0.1 9.5 Review Exercises and Sample Exam 2078 Chapter Sequences, Series, and the Binomial Theorem Given the terms of a geometric sequence, find a formula for the general term 69 4, 40, 400,… 70 −6, −30, −150,… 71 6, 27 , ,… 72 1, , ,… 25 73 a4 = −4 and a9 = 128 74 a2 = −1 and a5 = −64 75 a2 = − 76 a3 = 50 and a6 = −6,250 and a5 =− 625 16 77 Find all geometric means between a1 = −1 and a4 = 64 78 Find all geometric means between a3 = and a6 = 162 Calculate the indicated sum given the formula for the general term of a geometric sequence 79 an = 3(4)n−1 ; S 80 an = −5(3)n−1 ; S 10 81 an = (−2) n; S 14 82 an = (−3) n+1; S 12 83 84 an = 8( 12 ) n+2 an = ; S8 (−2) n+2; S 10 Evaluate ∑ 10 85 86 n=1 ∑ n=1 9.5 Review Exercises and Sample Exam − 3(−4) n (−2) n−1 2079 Chapter Sequences, Series, and the Binomial Theorem −3 ∑ (3) n=1 n ∞ 87 ∑ (5) n=1 n+1 ∞ 88 − ∑ ( 2) n=1 ∞ 89 − ∑ ( 2) n=1 n ∞ 90 n 91 After the first year of operation, the value of a company van was reported to be $40,000 Because of depreciation, after the second year of operation the van was reported to have a value of $32,000 and then $25,600 after the third year of operation Write a formula that gives the value of the van after the nth year of operation Use it to determine the value of the van after 10 years of operation 92 The number of cells in a culture of bacteria doubles every hours If 250 cells are initially present, write a sequence that shows the number of cells present after every 6-hour period for one day Write a formula that gives the number of cells after the nth 6-hour period 93 A ball bounces back to one-half of the height that it fell from If dropped from 32 feet, approximate the total distance the ball travels = 12,500(0.75) 94 A structured settlement yields an amount in dollars each year n according to the formula p n settlement? n−1 What is the total value of a 10-year Classify the sequence as arithmetic, geometric, or neither 95 4, 9, 14,… 96 6, 18, 54,… 97 −1, − , 0,… 98 10, 30, 60,… 99 0, 1, 8,… 100 −1, 9.5 Review Exercises and Sample Exam , − ,… 2080 Chapter Sequences, Series, and the Binomial Theorem Evaluate ∑ n2 ∑ n3 101 n=1 102 n=1 −4n + 5) ∑( 32 103 n=1 −2 ∑ (5) n=1 n−1 ∞ 104 (−3) n ∑ n=1 105 1 n− ∑ (4 2) n=1 46 106 ∑ 22 107 (3 − n) n=1 31 108 ∑ n=1 28 109 n=1 3(−1) n−1 ∑ 3(−1) n−1 n=1 31 111 ∑ ∑ 30 110 2n n=1 BINOMIAL THEOREM Evaluate 112 8! 9.5 Review Exercises and Sample Exam 2081 Chapter Sequences, Series, and the Binomial Theorem 113 11! 114 10! 2!6! 115 9!3! 8! 116 (n+3)! n! 117 (n−2)! (n+1)! Calculate the indicated binomial coefficient 118 (4) 119 (3) 120 10 ( ) 121 11 ( 10 ) 122 12 ( ) 123 n+1 (n − 1) 124 n (n − 2) Expand using the binomial theorem 125 (x + 7) 126 (x − 9) 127 (2y − 3) 9.5 Review Exercises and Sample Exam 2082 Chapter Sequences, Series, and the Binomial Theorem 128 129 130 (y + 4) (x + 2y) (3x − y) 131 (u − v) 132 (u + v) 133 134 5 2 (5x + 2y ) (x − 2y ) 9.5 Review Exercises and Sample Exam 2083 Chapter Sequences, Series, and the Binomial Theorem ANSWERS 2, 7, 12, 17, 22; a30 = 147 −10, −20, −30, −40, −50; a30 −1, 0, −1, 4, −9; a30 3, 11 x = 784 11 , , , ; a30 , 2x , 3x , = −300 4x = , 61 30 5x 11 2x , 4x , 8x , 16x , 32x 10 13 0, 5, 10, 15, 20 15 0, −3, 9, −30, 99 17 35 19 −5 21 −18 23 −36 25 29 27 135 29 6, 11, 16, 21, 26; an 31 5, 2, −1, −4, −7; an 33 − = 5n + = − 3n 3 15 , − , − , −3, − ; an 4 35 7, 7, 7, 7, 7; an n =7 37 an = 10n 39 an = − 3n 41 an = 3n − 43 an = =− n+3 45 7,140 9.5 Review Exercises and Sample Exam 2084 Chapter Sequences, Series, and the Binomial Theorem 47 33 49 1,674 51 1,661 53 420 55 1,842 57 30,625 59 1 , 0, − 3 61 $333,000 63 5, 10, 20, 40, 80; an 65 1, − = 5(2)n−1 27 81 , ,− , ;a 16 n = (− 32 ) 67 1.2, 0.24, 0.048, 0.0096, 0.00192; an 69 71 73 75 n−1 = 1.2(0.2) n−1 an = 4(10) n−1 an = 6( 34 ) n−1 a1 = (−2) n−1 an = −( 52 ) n−1 77 4, −16 79 4,095 81 16,383 83 255 128 85 2,516,580 87 −6 89 No sum 91 v n = 40,000(0.8) n−1 ; v 10 = $5,368.71 93 96 feet 9.5 Review Exercises and Sample Exam 2085 Chapter Sequences, Series, and the Binomial Theorem 95 Arithmetic; d =5 97 Arithmetic; d = 99 Neither 101 30 103 −1,952 105 1,640 107 −187 109 84 111 113 39,916,800 115 54 117 n(n+1)(n−1) 119 56 121 11 123 n(n+1) 125 x + 21x + 147x + 343 127 16y − 96y + 216y − 216y + 81 129 x + 10x y + 40x y + 80x y + 80xy + 32y 131 133 u − 6u v + 15u v − 20u v +15u v − 6uv + v 625x + 1,000x y + 600x y + 160x y + 16y 9.5 Review Exercises and Sample Exam 2086 Chapter Sequences, Series, and the Binomial Theorem SAMPLE EXAM Find the first terms of the sequence an = 6n − 15 an = 5(−4) n−2 an = an = (−1) n−1 x 2n n−1 2n−1 Find the indicated partial sum an = (n − 1) n 2; S ∑ k=1 (−1) k k−2 Classify the sequence as arithmetic, geometric, or neither −1, − , −2,… 1, −6, 36,… 10 3 , − , ,… 1 , , ,… Given the terms of an arithmetic sequence, find a formula for the general term 11 10, 5, 0, −5, −10,… 12 a4 = − and a9 =2 Given the terms of a geometric sequence, find a formula for the general term 1 , − , −2, −8, −32,… 13 − 14 a3 = and a8 = −32 Calculate the indicated sum 9.5 Review Exercises and Sample Exam 2087 Chapter Sequences, Series, and the Binomial Theorem 15 an = − n; S 44 16 an = (−2) n+2; S 12 − ∑ ( 2) n=1 n−1 ∞ 17 ∑( 100 18 2n − n=1 2) Evaluate 19 14! 10!6! 20 (7) 21 Determine the sum of the first 48 positive odd integers 22 The first row of seating in a theater consists of 14 seats Each successive row consists of two more seats than the previous row If there are 22 rows, how many total seats are there in the theater? 23 A ball bounces back to one-third of the height that it fell from If dropped from 27 feet, approximate the total distance the ball travels Expand using the binomial theorem 24 25 (x − 5y) (3a + b ) 9.5 Review Exercises and Sample Exam 2088 Chapter Sequences, Series, and the Binomial Theorem ANSWERS −9, −3, 3, 9, 15 0, , , , 70 Arithmetic Geometric 11 an = 15 − 5n 13 an = − (4)n−1 15 −770 17 19 1,001 30 21 2,304 23 54 feet 25 9.5 Review Exercises and Sample Exam 243a5 + 405a4 b + 270a3 b +90a2 b + 15ab + b 10 2089 ... Intermediate Algebra is the second part of a two-part course in Algebra Written in a clear and concise manner, it carefully builds on the basics learned in Elementary Algebra and introduces the more advanced. .. 1: Algebra Fundamentals Review of Real Numbers and Absolute Value Operations with Real Numbers 35 Square and Cube Roots of Real Numbers 68 Algebraic...This is the book Advanced Algebra (v 1.0) This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/