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The Art of Problem Solving - Introduction to Algebra
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Academic Year Mathematics
In Algebra 1, students engage with various expressions from an algebraic and geometric perspective. They learn to solve linear and quadratic equations as well as how to represent expressions in the Cartesian plane. Students are imbued with a deep understanding of the general concept behind functions and are introduced to several important types of functions. Students are also introduced to the rich field of complex numbers. As with previous courses, we continue to emphasize challenging word problems to train students how to apply their new tools. Advanced problems from contests such as MATHCOUNTS, AMC 8, and AMC 10 are included throughout the course.
Homework and exams
Students should expect to spend 1-3 hours on homework every week. Homework will be assigned and completed through the class website, where students receive immediate feedback on their work. Students will take three in-class exams throughout the course.
The textbook for the course is Introduction to Algebra by Art of Problem Solving.
Hardcopy and lifetime digital access to the online version of the textbook are included in the tuition and fees of this course.
Below are examples of some of the types of problems that students will encounter in our Algebra 1 course.
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Intermediate Algebra: Art of Problem Solving PDF
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Table of contents.
How to Use This Book Acknowledgments Contents 1 Basic Techniques for Solving Equations 1.1 Isolation 1.2 Substitution 1.3 Elimination 1.4 Larger Systems of Linear Equations 1.5 Summary 2 Functions Review 2.1 Function Basics 2.2 Graphing Functions 2.3 Composition 2.4 Inverse Functions 2.5 Summary 3 Complex Numbers 3.1 Arithmetic of Complex Numbers 3.2 The Complex Plane 3.3 Real and Imaginary Parts 3.4 Graphing in the Complex Plane 3.5 Summary 4 Quadratics 4.1 Factoring Quadratics 4.2 Relating Roots and Coefficients 4.3 Completing the Square 4.4 The Discriminant 4.5 Quadratic Inequalities 4.6 Summary 5 Conics 5.1 Parabolas 5.2 Problem Solving With Parabolas 5.3 Maxima and Minima of Quadratics 5.4 Circles 5.5 Ellipses 5.6 Hyperbolas 5.7 Summary 6 Polynomial Division 6.1 Polynomial Review 6.2 Introduction to Polynomial Division 6.3 Synthetic Division 6.4 The Remainder Theorem 6.5 Summary 7 Polynomial Roots Part I 7.1 The Factor Theorem 7.2 Integer Roots 7.3 Rational Roots 7.4 Bounds 7.5 Graphing and the Fundamental Theorem of Algebra 7.6 Algebraic Applications of the Fundamental Theorem 7.7 Summary 8 Polynomial Roots Part II 8.1 Irrational Roots 8.2 Nonreal Roots 8.3 Vieta's Formulas 8.4 Using Roots to Make Equations 8.5 Summary 9 Factoring Multivariable Polynomials 9.1 Grouping 9.2 Sums and Differences of Powers 9.3 The Factor Theorem for Multivariable Polynomials 9.4 Summary 10 Sequences and Series 10.1 Arithmetic Sequences 10.2 Arithmetic Series 10.3 Geometric Sequences 10.4 Geometric Series 10.5 Sequence, Summation, and Product Notation 10.6 Nested Sums and Products 10.7 Summary 11 Identities, Manipulations, and Induction 11.1 Brute Force 11.2 Ratios 11.3 Induction 11.4 Binomial Theorem 11.5 Summary 12 Inequalities 12.1 Manipulating Inequalities 12.2 The Trivial Inequality 12.3 AM-GM Inequality with Two Variables 12.4 AM-GM with More Variables 12.5 The Cauchy-Schwarz Inequality 12.6 Maxima and Minima 12.7 Summary 13 Exponents and Logarithms 13.1 Exponential Function Basics 13.2 Introduction to Logarithms 13.3 Logarithmic Identities 13.4 Using Logarithm Identities 13.5 Switching Between Logs and Exponents 13.6 Natural Logarithms and Exponential Decay 13.7 Summary 14 Radicals 14.1 Raising Radicals to Powers 14.2 Evaluating Expressions With Radicals 14.3 Radical Conjugates 14.4 Summary 15 Special Classes of Functions 15.1 Rational Functions and Their Graphs 15.2 Rational Function Equations and Inequalities 15.3 Even and Odd Functions 15.4 Monotonic Functions 15.5 Summary 16 Piecewise Defined Functions 16.1 Introduction to Piecewise Defined Functions 16.2 Absolute Value 16.3 Graphing Absolute Value 16.4 Floor and Ceiling 16.5 Problem Solving with the Floor Function 16.6 Summary 17 More Sequences and Series 17.1 Algebra of Recursive Sequences 17.2 Telescoping 17.3 Sums of Polynomial Series 17.4 Arithmetico-Geometric Series 17.5 Finite Differences 17.6 Summary 18 More Inequalities 18.1 Mean Inequality Chain 18.2 The Rearrangement Inequality 18.3 When Formulas Fail 18.4 Summary 19 Functional Equations 19.1 Finding Values 19.2 Finding Functions with Substitution 19.3 Separation 19.4 Cyclic Functions 19.5 Summary 20 Some Advanced Strategies 20.1 Symmetry 20.2 Substitution for Simplification 20.3 Method of Undetermined Coefficients 20.4 Constructing Polynomials From Roots 20.5 Common Divisors of Polynomials 20.6 Symmetric Sums Revisited 20.7 Summary References Hints to Selected Problems Index