College Algebra with Intermediate Algebra: A Blended Course, 1st edition

To the instructor: As you know from your experience teaching both Intermediate Algebra and College Algebra courses, there is a great deal of repetition of topics in the two courses. In fact, over 60% of the topics covered in a College Algebra textbook are also covered in an Intermediate Algebra text. This book is intended to eliminate this repetition and consequently to create a course that makes better use of students’ time and resources.

ANNOTATED TABLE OF CONTENTS

Chapter R Review of Basic Algebra

Part 1 Operations

R.1 The Set of Real Numbers

R.2 Operations with Real Numbers

R.3 Exponential Notation and Order of Operations

Part 2 Manipulations

R.4 Introduction to Algebraic Expressions

R.5 Equivalent Algebraic Expressions

R.6 Simplifying Algebraic Expressions

R.7 Properties of Exponents and Scientific Notation

1 Solving Linear Equations and Inequalities

The topics in this chapter are often taught in Intermediate Algebra and then again in College Algebra.  Here, the repetition is eliminated.  

1.1 Solving Equations

1.2 Formulas and Applications

1.3 Applications and Problem Solving

1.4 Sets, Inequalities, and Interval Notation

1.5 Intersections, Unions, and Compound Inequalities

1.6 Absolute-Value Equations and Inequalities

2 Graphs, Functions, and Applications

We believe that functions and graphing should be introduced early and continue as a thread that runs through the course. This allows the instructor the opportunity to use the visual element of graphing to show students how solutions of equations, zeros of functions, and x-intercepts of graphs are related.  This chapter blends function topics from Intermediate Algebra and College Algebra.

2.1 Graphs of Equations

2.2 Functions and Graphs

2.3 Finding Domain and Range

2.4 The Algebra of Functions

2.5 Linear Functions: Graphs and Slope

2.6 More on Graphing Linear Equations

2.7 Finding Equations of Lines; Applications

3 Systems of Equations

This chapter covers systems of equations, a topic that is often taught in both Intermediate Algebra and College Algebra. This is another instance in which repetition of topics is eliminated.  In addition, it includes linear programming, a topic that is traditionally in College Algebra but not in Intermediate Algebra.

3.1 Systems of Equations in Two Variables

3.2 Solving by Substitution

3.3 Solving by Elimination

3.4 Solving Applied Problems: Two Equations

3.5 Systems of Equations in Three Variables

3.6 Solving Applied Problems: Three Equations

3.7 Systems of Inequalities and Linear Programming

4 Polynomials and Polynomial Functions

This chapter provides thorough coverage of polynomials and polynomial functions, thus laying a solid foundation for the more advanced College Algebra coverage of these topics in Chapter 8.

4.1 Introduction to Polynomials and Polynomial Functions

4.2 Multiplication of Polynomials

4.3 Introduction to Factoring

4.4 Factoring Trinomials: x2 + bx + c

4.5 Factoring Trinomials: ax2 + bx + c, a not equal to 1

4.6 Special Factoring

4.7 Factoring: A General Strategy

4.8 Applications of Polynomial Equations and Functions

5 Rational Expressions, Equations, and Functions

In this chapter, the student gets a solid basic coverage of rational expressions, rational equations, and rational functions as well as coverage of the difference quotient from College Algebra.  This paves the way for studying more advanced College Algebra coverage of many of these topics in Chapter 8.

5.1 Rational Expressions and Functions: Multiplying, Dividing, and Simplifying

5.2 LCMs, LCDs, Addition, and Subtraction

5.3 Division of Polynomials

5.4 Complex Rational Expressions

5.5 Solving Rational Equations

5.6 Applications and Proportions

5.7 Formulas and Applications

5.8 Variation and Applications

6 Radical Expressions, Equations, and Functions

This chapter covers traditional Intermediate Algebra topics. In addition, now that the student has studied linear functions, polynomial functions of degree two and higher, rational functions, and radical functions, we have included the College Algebra topics of increasing functions, decreasing functions, and piecewise-defined functions. We weave together topics that are covered early in College Algebra and topics from Intermediate Algebra.  (Note that complex numbers are covered in Chapter 7 along with quadratic equations rather than in this chapter.)

6.1 Radical Expressions and Functions

6.2 Rational Numbers as Exponents

6.3 Simplifying Radical Expressions

6.4 Addition, Subtraction, and More Multiplication

6.5 More on Division of Radical Expressions

6.6 Solving Radical Equations

6.7 Applications Involving Powers and Roots

6.8 Increasing, Decreasing, and Piecewise Functions; Applications

7 Quadratic Functions and Equations

This chapter blends traditional Intermediate Algebra coverage of quadratic equations with additional, more advanced topics that are traditionally associated with a College Algebra course.

7.1 Symmetry

7.2 Transformations

7.3 The Complex Numbers

7.4 Quadratic Equations, Functions, Zeros, and Models

7.5 Analyzing Graphs of Quadratic Functions

8 Polynomial Functions and Rational Functions

Here, College Algebra topics regarding polynomial functions and rational functions are built on the solid foundation that was laid in Chapters 4 and 5.

8.1 Polynomial Functions and Models

8.2 Graphing Polynomials Functions

8.3 Polynomial Division; The Remainder Theorem and the Factor Theorem

8.4 Theorems about Zeros of Polynomial Functions

8.5 Rational Functions

8.6 Polynomial Inequalities and Rational Inequalities

9 Exponential Functions and Logarithmic Functions

Traditionally, there is little difference in the Intermediate Algebra coverage vs. College Algebra coverage of the topics in this chapter.  This is an excellent opportunity to eliminate repetition and thus streamline students’ investment of time and resources. Although inverse functions are often introduced earlier in a College Algebra course, we introduce them here so that they will be fresh in the students’ minds when they are used to show the relationship between exponential functions and logarithmic functions.

9.1  The Composition of Functions

9.2 Inverse Functions

9.3 Exponential Functions and Graphs

9.4 Logarithmic Functions and Graphs

9.5 Properties of Logarithmic Functions

9.6 Solving Exponential Equations and Logarithmic Equations

9.7 Applications and Models: Growth and Decay; Compound Interest

10 Matrices

These are the matrix topics that are traditionally taught in a College Algebra course, often along with systems of equations. Here they are presented in a separate chapter so that systems of equations could be covered earlier in the course with the more traditional Intermediate Algebra topics.

10.1 Matrices and Systems of Equations

10.2 Matrix Operations

10.3 Inverses of Matrices

10.4 Determinants and Cramer’s Rule

11 Conic Sections

In the topics of conic sections and nonlinear systems of equations we find another area of considerable overlap in Intermediate Algebra and College Algebra.  This is another example of how repetition is eliminated.

11.1 The Parabola

11.2 The Circle and the Ellipse

11.3 The Hyperbola

11.4 Nonlinear Systems of Equations and Inequalities

12 Sequences, Series, and Combinatorics

These are traditional College Algebra topics.

12.1 Sequences and Series

12.2 Arithmetic Sequences and Series

12.3 Geometric Sequences and Series

12.4 Mathematical Induction

12.5 Combinatorics: Permutations

12.6 Combinatorics: Combinations

12.7 The Binomial Theorem

12.8 Probability

Appendix

This is a College Algebra topic that is often considered to be optional, so we have presented it in an appendix.