ALGEBRA and TRIGONOMETRY PAUL SISSON Annotated Instructor’s Edition - Not for SaleEditors: Danielle C. Bess, Daniel Breuer, S. Rebecca Johnson, Claudia Vance Assistant Editors: Sarah L. Allen, Allison Conger, Marvin Glover, Lisa Hinton Index: Barbara Miller Manager of Math Content Development: Blair Dunivan Creative Services Manager: Trudy Tronco Designers: Lizbeth Mendoza, Patrick Thompson, Joel Travis, Joshua A. Walker Composition Assistance: Quant Systems India Pvt. Ltd. Courseware Developers: Vince Cellini, Adam Flaherty Technology Assistant: Kyle Gilstrap Cover Artwork: George Hart George Hart is a sculptor and applied mathematician who demonstrates how mathematics is cool and creative in ways you might not have expected. Whether he is slicing a bagel into two linked halves or leading hundreds of participants in an intricate geometric sculpture barn raising, he always finds original ways to share the beauty of mathematical thinking. Hart’s career includes eight years as a professor at Columbia University, fifteen years as a Research Professor at Stony Brook University, and five years cofounding the Museum of Mathematics in New York City. Now a full-time artist and consultant, he also makes videos that show the fun and creative sides of mathematics. See http://georgehart.com for examples of his work. A division of Quant Systems, Inc. 546 Long Point Road Mount Pleasant, SC 29464 Copyright © 2023 by Hawkes Learning / Quant Systems, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written consent of the publisher. Library of Congress Control Number 2022933310 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN: 978-1-64277-528-0 AIE ISBN: 978-1-64277-531-0Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Fundamental Concepts of Algebra 1 .1 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Common Subsets of Real Numbers w The Real Number Line w Order on the Real Number Line w Set-Builder Notation and Interval Notation w Absolute Value and Distance on the Real Number Line w Working with Repeating Decimals 1 .2The Arithmetic of Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 Components and Terminology of Algebraic Expressions w The Field Properties and Their Use in Algebra w The Order of Mathematical Operations w Basic Set Operations and Venn Diagrams 1 .3 Properties of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 Natural Number Exponents w Integer Exponents w Properties of Exponents w Scientific Notation w Working with Geometric Formulas 1 .4 Properties of Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 Radical Notation w Simplifying Radical Expressions w Combining Radical Expressions w Rational Number Exponents 1 .5 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54 The Terminology of Polynomial Expressions w Basic Operations with Polynomials 1 .6 Factoring Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .60 Greatest Common Factor (GCF) w Factoring by Grouping w Factoring Special Binomials w Factoring Trinomials w Factoring Expressions Containing Noninteger Rational Exponents 1 .7 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 Simplifying Rational Expressions w Combining Rational Expressions w Simplifying Complex Rational Expressions 1 .8 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76 The Imaginary Unit i and Its Properties w Basic Operations with Complex Numbers w Roots and Complex Numbers Chapter 1 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84 Chapter 1 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85 2 Equations and Inequalities in One Variable 2 .1 Linear Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 Equations and the Meaning of Solutions w Solving Linear Equations w Solving Linear Absolute Value Equations w Solving Linear Equations for One Variable w Applications of Linear Equations 2 .2Linear Inequalities in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104 Solving Linear Inequalities w Solving Double Linear Inequalities w Solving Linear Absolute Value Inequalities w Applications of Linear Inequalities 2 .3Quadratic Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 Solving Quadratic Equations by Factoring w Solving “Perfect Square” Quadratic Equations w Solving Quadratic Equations by Completing the Square w The Quadratic Formula w Applications of Quadratic Equationsiv Table of Contents 2 .4Polynomial and Polynomial-Like Equations in One Variable . . . . . . . . . . . . . . . . . . . . . .128 Solving Quadratic-Like Equations w Solving General Polynomial Equations by Factoring w Solving Polynomial-Like Equations by Factoring 2 .5Rational Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134 Solving Rational Equations w Applications of Rational Equations 2 .6Radical Equations in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140 Solving Radical Equations w Solving Equations with Positive Rational Exponents w Solving Equations for One Variable Chapter 2 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147 Chapter 2 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148 3 Equations and Inequalities in Two Variables 3 .1 The Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 The Cartesian Coordinate System w Graphing Equations w The Distance and Midpoint Formulas 3 .2 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168 The Standard Form of the Equation of a Circle w Graphing Circles 3 .3 Linear Equations in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174 Recognizing Linear Equations in Two Variables w Intercepts of the Coordinate Axes w Horizontal and Vertical Lines 3 .4 Slope and Forms of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 The Slope of a Line w The Slope-Intercept Form of the Equation of a Line w The Point-Slope Form of the Equation of a Line 3 .5 Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193 Slopes of Parallel Lines w Slopes of Perpendicular Lines 3 .6 Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201 Graphing Linear Inequalities w Graphing Linear Inequalities Joined by “And” or “Or” w Graphing Linear Absolute Value Inequalities Chapter 3 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .211 Chapter 3 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212 4 Relations, Functions, and Their Graphs 4 .1 Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .219 Relations, Domain, and Range w Functions and the Vertical Line Test w Function Notation and Function Evaluation w Implied Domain of a Function 4 .2 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237 Linear Functions and Their Graphs w Linear Regression 4 .3 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252 Quadratic Functions and Their Graphs w Quadratic Regression w Maximization/Minimization Problems 4 .4 Other Common Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .271 Power Functions of the Form axn w Power Functions of the Form ax n - w Power Functions of the Form axn 1 w The Absolute Value Function w The Greatest Integer Function w Piecewise-Defined FunctionsTable of Contents v 4 .5 Variation and Multivariable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .281 Direct Variation w Inverse Variation w Joint Variation w Multivariable Functions 4 .6 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .289 Constructing Mathematical Models w Interpolation and Extrapolation Chapter 4 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303 Chapter 4 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304 5 Working with Functions 5 .1 Transformations of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313 Shifting Graphs Vertically and Horizontally w Reflecting Graphs w Stretching Graphs Vertically and Horizontally w Order of Transformations 5 .2 Properties of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .326 Symmetry of Functions and Equations w Intervals of Monotonicity w Local Extrema w Average Rate of Change 5 .3 Combining Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .341 Combining Functions Arithmetically w Composing Functions w Decomposing Functions w Recursive Graphics 5 .4 Inverses of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353 Inverses of Relations w Inverse Functions and the Horizontal Line Test w Finding the Inverse of a Function Chapter 5 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .365 Chapter 5 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .366 6 Polynomial and Rational Functions 6 .1 Polynomial Functions and Polynomial Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .373 Zeros of Polynomial Functions and Solutions of Polynomial Equations w Graphing Factored Polynomial Functions w Solving Polynomial Inequalities 6 .2 Polynomial Division and the Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .389 The Division Algorithm and the Remainder Theorem w Polynomial Long Division w Synthetic Division w Constructing Polynomials with Given Zeros 6 .3 Locating Real Zeros of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .401 The Rational Zero Theorem w Descartes’ Rule of Signs w Bounds of Real Zeros w The Intermediate Value Theorem 6 .4 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .412 The Fundamental Theorem of Algebra w Multiple Zeros and Their Geometric Meaning w Conjugate Pairs of Zeros w Summary of Polynomial Methods 6 .5 Rational Functions and Rational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .423 Characteristics of Rational Functions w Vertical Asymptotes w Horizontal and Oblique Asymptotes w Graphing Rational Functions w Solving Rational Inequalities Chapter 6 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .441 Chapter 6 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .442vi Table of Contents 7 Exponential and Logarithmic Functions 7 .1 Exponential Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .449 Characteristics of Exponential Functions w Graphing Exponential Functions w Solving Elementary Exponential Equations 7 .2 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .458 Models of Population Growth w Models of Radioactive Decay w Compound Interest and the Number e w Exponential Regression 7 .3 Logarithmic Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .475 Characteristics of Logarithmic Functions w Graphing Logarithmic Functions w Evaluating Elementary Logarithmic Expressions w Solving Elementary Logarithmic Equations w Common and Natural Logarithms 7 .4 Logarithmic Properties and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .487 Properties of Logarithms w The Change of Base Formula w Applications of Logarithmic Functions w Logarithmic Regression 7 .5 Exponential and Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .501 Converting between Exponential and Logarithmic Forms w Applications of Exponential and Logarithmic Equations w Analysis of a Stock Market Investment Chapter 7 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .513 Chapter 7 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .514 8 Trigonometric Functions 8 .1 Radian and Degree Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .521 The Unit Circle and Angle Measure w Converting between Degrees and Radians w The Pythagorean Theorem and Commonly Encountered Angles w Arc Length and Angular Speed w Area of a Circular Sector 8 .2 Trigonometric Functions and Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .535 The Trigonometric Functions w Evaluating Trigonometric Functions w Applications of Trigonometric Functions 8 .3 Trigonometric Functions and the Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .548 Extending the Domains of the Trigonometric Functions w Evaluating Trigonometric Functions Using Reference Angles w Relationships between Trigonometric Functions 8 .4 Graphs of Sine and Cosine Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .561 Graphing Sine and Cosine Functions w Periodicity and Symmetry w Amplitude, Frequency, and Phase Shifts w Simple Harmonic Motion w Damped Harmonic Motion 8 .5 Graphs of Other Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .577 Graphing Tangent and Cotangent Functions w Graphing Secant and Cosecant Functions 8 .6 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 The Inverse Trigonometric Functions w Evaluating Inverse Trigonometric Functions w Applications of Inverse Trigonometric Functions Chapter 8 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .593 Chapter 8 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .594Table of Contents vii 9 Trigonometric Identities and Equations 9 .1 Fundamental Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .601 Fundamental Trigonometric Identities w Simplifying Trigonometric Expressions w Verifying Trigonometric Identities w Trigonometric Substitutions 9 .2 Sum and Difference Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .609 Sum and Difference Identities w Using Sum and Difference Identities for Exact Evaluation w Using Sum and Difference Identities for Verification and Simplification 9 .3 Product-Sum Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .620 Double-Angle Identities w Power-Reducing Identities w Half-Angle Identities w Product-to-Sum and Sum-to-Product Identities 9 .4 Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .629 Solving Trigonometric Equations Using Algebraic Techniques w Solving Trigonometric Equations Using Inverse Trigonometric Functions Chapter 9 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .639 Chapter 9 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .640 10 Additional Topics in Trigonometry 10 .1 The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .647 The Law of Sines w Applications of the Law of Sines w The Law of Sines and the Area of a Triangle 10 .2 The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .656 The Law of Cosines w Applications of the Law of Cosines w Heron’s Formula for the Area of a Triangle 10 .3 Polar Coordinates and Polar Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .663 The Polar Coordinate System w Converting between Polar and Cartesian Coordinates w The Form of Polar Equations w Graphing Polar Equations 10 .4 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .676 Applications of Parametric Equations w Graphing Parametric Equations by Eliminating the Parameter w Constructing Parametric Equations to Describe a Graph 10 .5 Trigonometric Form of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .687 The Complex Plane w Complex Numbers in Trigonometric Form w Multiplying and Dividing Complex Numbers w Powers of Complex Numbers w Roots of Complex Numbers 10 .6 Vectors in the Cartesian Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .697 Vector Terminology w Basic Vector Operations w Component Form of a Vector w Applications of Vectors 10 .7 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .707 Properties of the Dot Product w Projections of Vectors w Applications of the Dot Product 10 .8 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .716 The Hyperbolic Functions w Hyperbolic Identities w The Inverse Hyperbolic Functions Chapter 10 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .725 Chapter 10 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .726viii Table of Contents 11 Conic Sections 11 .1 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .735 Overview of Conic Sections w The Standard Form of the Equation of an Ellipse w Applications of Ellipses 11 .2 Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .748 The Standard Form of the Equation of a Parabola (as a Conic Section) w Applications of Parabolas 11 .3 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .757 The Standard Form of the Equation of a Hyperbola w Applications of Hyperbolas 11 .4 Rotation of Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .768 Rotation Relations w Rotation Invariants 11 .5 Polar Equations of Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .778 The Focus and Directrix of a Conic Section w Graphing Polar Equations of Conic Sections Chapter 11 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .788 Chapter 11 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .789 12 Systems of Equations and Inequalities 12 .1 Solving Systems of Linear Equations by Substitution and Elimination . . . . . . . . . . . . . .797 Systems of Linear Equations w Solving Systems of Linear Equations by Substitution w Solving Systems of Linear Equations by Elimination w Systems of Linear Equations in Three or More Variables w Applications of Systems of Linear Equations 12 .2 Matrix Notation and Gauss-Jordan Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .810 Systems of Linear Equations, Matrices, and Augmented Matrices w Gaussian Elimination and Row Echelon Form w Gauss-Jordan Elimination and Reduced Row Echelon Form 12 .3 Determinants and Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .824 Evaluating Determinants w Solving Systems of Linear Equations Using Cramer’s Rule 12 .4 Basic Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .837 Matrix Addition w Scalar Multiplication w Matrix Multiplication w Transition Matrices 12 .5 Inverses of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .850 The Matrix Form of a System of Linear Equations w Finding the Inverse of a Matrix w Solving Systems of Linear Equations Using Matrix Inverses 12 .6 Partial Fraction Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .862 The Pattern of Partial Fraction Decompositions w Completing the Partial Fraction Decomposition Process 12 .7 Systems of Linear Inequalities and Linear Programming . . . . . . . . . . . . . . . . . . . . . . . .871 Systems of Linear Inequalities w Planar Feasible Regions w Linear Programming in Two Variables 12 .8 Systems of Nonlinear Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .881 Solving Systems of Nonlinear Equations by Graphing w Solving Systems of Nonlinear Equations Algebraically w Solving Systems of Nonlinear Inequalities Chapter 12 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .892 Chapter 12 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .893Table of Contents ix 13 Sequences, Series, Combinatorics, and Probability 13 .1 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .903 Recursively and Explicitly Defined Sequences w Summation Notation and Formulas w Partial Sums and Series w Fibonacci Sequences 13 .2 Arithmetic Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 Arithmetic Sequences and Series w The Formula for the General Term of an Arithmetic Sequence w Evaluating Partial Sums of Arithmetic Sequences 13 .3 Geometric Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .926 Geometric Sequences w The Formula for the General Term of a Geometric Sequence w Evaluating Partial Sums of Geometric Sequences w Evaluating Infinite Geometric Series w Zeno’s Paradoxes 13 .4 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .940 The Role of Induction w Proofs by Mathematical Induction 13 .5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .950 The Multiplication Principle of Counting w Permutations w Combinations w The Binomial Theorem w The Multinomial Theorem 13 .6 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .967 The Language of Probability w Computing Probabilities Using Combinatorics w Unions, Intersections, and Independent Events Chapter 13 Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .978 Chapter 13 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .979 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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