College Algebra

The topics are Algebra Review, Graphs, Functions and their Graphs, Equations and Inequalities, Polynomial and Rational Functions, Exponential and Logarithmic Functions, Systems of Equations, Sequence, Series and Binomial Theorem.

UNIT 0 - BASICS: Remembrance of Things Past

Lecture 1: NUMBERS
Sets of Objects [4 min.]
Natural Numbers [7 min.]
Integers [5 min.]
Rational Numbers [13 min.]
Irrational Numbers [2 min.]
Real Numbers [1 min.]
The Real Line: From Numbers to Points [4 min.]
Real Numbers are Ordered [19 min,]
The Real Line: Distance Between Points [8 min.]
Real Numbers as Decimals [11 min.]

Lecture 2: THE LANGUAGE OF MATHEMATICS
Learning to Read Mathematics [7 min.]
Some Symbols of Algebra [9 min.]
Nouns, Pronouns, & The Main Verb of Algebra [4 min.]
Theorems, Corollaries, Lemmas, and All That [7 min.]

Lecture 3: THE POWERS THAT BE: EXPONENTS
Integer Exponents [12 min.]
Operations with Integer Exponents [31 min.]
Square Roots: A Pair Of Equal Factors [13 min.]
Nth Roots & Rational Exponents [12 min.]
Operations with Rational Exponents [21 min.]

Lecture 4 Part 1: POLYNOMIAL EXPRESSIONS
What is a Polynomial? [18 min.]
Adding & Subtracting Polynomials [3 min.]
Multiplying Polynomials [19 min.]
A Common Error [5 min.]
Handy Polynomial Products [15 min.]
Un-Multiplying (Factoring) Polynomials [28 min.]
Completing A Perfect Square [10 min.]

Lecture 4 Part 2: POLYNOMIAL EXPRESSION
Dividing Polynomials: Rational Expressions [24 min.]
The Art of Simplification [14 min.]
Solving Some Polynomial & Rational Equations [46 min.]

Lecture 5: MORE NUMBERS and GEOMETRY
Beyond Real Numbers: Complex Numbers [19 min.]
Some Area Formulas [8 min.]
The Pythagorean Theorem & A Visual Proof [12 min.]

UNIT 1 - GRAPHS

Lecture 6: GRAPHS
Rectangular Coordinates: Geometry Meets Algebra [11 min.]
Distance Between Points [18 min.]
Midpoint of a Line Segment [7 min.]
Your Graphing Device [17 min.]

Lecture 7: GRAPHS
Graphs of Equations [25 min.]
Intercepts: Crossing the Axes [12 min.]
Symmetry of Graphs [14 min.]
Lines: Defining “Slope” [34 min.]

Lecture 8: GRAPHS
Lines & Their Equations [24 min.]
Parallel & Perpendicular Lines [26 min.]
Circles & Their Equations [28 min.]
Some Exercises Explained [16 min.]

UNIT 2 - FUNCTIONS AND THEIR GRAPHS

Lecture 9: FUNCTIONS AND THEIR GRAPHS
Function: The Central Idea of Mathematics [26 min.]
The Language & Notation of Functions [20 min.]
More On Domains [19 min.]
Function Notation Practice [16 min.]

Lecture 10: FUNCTIONS AND THEIR GRAPHS
Visualizing Functions: Graphs of (x, f(x)) Pairs [26 min.]
Increasing & Decreasing Functions [23 min.]
Local Maximums & Local Minimums [18 min.]
Even & Odd Functions [20 min.]

Lecture 11: FUNCTIONS AND THEIR GRAPHS
A Library of Important Functions [20 min.]
Piecewise Defined Functions [19 min.]
Some Exercises Explained [11 min.]

Lecture 12: FUNCTIONS AND THEIR GRAPHS
Graphing Techniques: Vertical & Horizontal Shifts [28 min.]
Graphing Techniques: Compressions & Stretches [27 min.]
Graphing Techniques: Reflections Across the Axes [17 min.]
Putting It All Together: Moving The Graphs of Functions Around The Plane [36 min.]

Lecture 13: FUNCTIONS AND THEIR GRAPHS
The Algebra of Functions [16 min.]
A New Operation Unique To Functions: Composition [24 min.]

Lecture 14: FUNCTIONS AND THEIR GRAPHS
Mathematical Models of Real World Problems: Constructing Functions [55 min.]

UNIT 3 - EQUATIONS & INEQUALITIES

Lecture 15: EQUATIONS IN 1 VARIABLE
Solving Equations (Approximately) with a Graphing Device & The Intermediate Value Theorem [44 min.]
Solving Linear Equations: The “Linear Formula” or, Graphing [23 min.]
Solving Non-Linear Equations That Lead To Linear Equations [13 min.]

Lecture 16: EQUATIONS IN 1 VARIABLE
Solving Quadratic Equations: Factoring, or, Graphing [11 min.]
A Complex Reminder & The Principal Square Root of a Negative Number [12 min.]
Solving Quadratic Equations: The “Quadratic Formula” & The Discriminant or, Graphing [50 min.]

Lecture 17: EQUATIONS IN 1 VARIABLE]
Some Linear & Quadratic Equation Exercises Explained [24 min.]

Lecture 18: EQUATIONS IN 1 VARIABLE
Setting Up Equations: More Mathematical Models [65 min.]

Lecture 19: EQUATIONS IN 1 VARIABLE]
Solving “Radical” Equations [14 min.]
Solving Equations “Quadratic in Form” [15 min.]
Solving Factorable Equations [7 min.]

Lecture 20: INEQUALITIES IN 1 VARIABLE
Properties of Inequalities [20 min.]
Solving Inequalities In General [17 min.]
Solving Linear Inequalities [19 min.]

Lecture 21: INEQUALITIES IN 1 VARIABLE
Solving Quadratic Inequalities [17 min.]
Solving Higher-Degree Polynomial Inequalities [14 min.]
Solving Rational Inequalities [27 min.]

Lecture 22: EQUATIONS & INEQUALITIES IN 1 VARIABLE
When Absolute Value Appears: Equations [12 min.]
When Absolute Value Appears: Inequalities [17 min.]
More Exercises Explained [15 min.]

UNIT 4 - POLYNOMIAL & RATIONAL FUNCTIONS

Lecture 23: POLYNOMIAL FUNCTIONS
Degree 2: Quadratic Functions [19 min.]
Graphing Quadratic Functions [39 min.]
Quadratic Functions As Mathematical Models [25 min.]

Lecture 24: POLYNOMIAL FUNCTIONS
Degree “n”: General Polynomial Functions [8 min.]
Special Case: Power Functions & Their Graphs [18 min.]
Graphing General Polynomial Functions: Zeros, Multiplicity, Turning Points & End Behavior [40 min.]

Lecture 25: LOCATING THE ZEROS OF A POLYNOMIAL FUNCTION
How Many Zeros Are There? [12 min.]
How Many Zeros Are Real? [13 min.]
How Many Real Zeros Are Positive Negative? [16 min.]
Where (On What Interval) Are All Real Zeros? [22 min.]
How Can You Guess The Location of Real Zeros? [4 min.]
How Can You Reduce The Number of Real Zeros? [21 min.]

Lecture 26: LOCATING THE ZEROS OF A POLYNOMIAL FUNCTION
Strategy & Tools: A Practical Checklist [14 min.]
Some Polynomial Exercises Explained [22 min.]

Lecture 27: RATIONAL FUNCTIONS
General Rational Functions [14 min.]
What Is An Asymptote? [17 min.]
Finding Asymptotes of Rational Functions [35 min.]
Graphing Rational Functions [19 min.]

UNIT 5 - EXPONENTIAL & LOGARITHMIC FUNCTIONS

Lecture 28: EXPONENTIAL FUNCTIONS
One-To-One Functions [14 min.]
Exponential Functions & Their Graphs [26 min.]
The Natural Exponential Function [15 min.]

Lecture 29: LOGARITHMIC FUNCTIONS
Inverse Functions [23 min.]
Logarithmic Functions & Their Graphs [24 min.]
The Natural Logarithmic Function [4 min.]

Lecture 30: LOGARITHMIC FUNCTIONS
Properties of Logarithms [17 min.]
All Logarithms are Natural (or Common)! [15 min.]
Solving Logarithmic Equations [20 min.]
Logarithmic Models: Sound (Loudness) & Fury (Earthquakes) [16 min.]

Lecture 31: EXPONENTIAL FUNCTIONS (con.)
Solving Exponential Equations [15 min.]
Exponential Models: Compounded Interest, and Growth & Decay [32 min.]

UNIT 6 - SYSTEMS OF EQUATIONS

Lecture 32: SYSTEMS OF LINEAR EQUATIONS
Systems of Linear Equations in General [19 min.]
Solving A System of 2 (or 3) Linear Equations in 2 (or 3) Variables: Substitution [20 min.]
Solving A System of 2 (or 3) Linear Equations in 2 (or 3) Variables: Elimination [36 min.]

Lecture 33: SYSTEMS OF LINEAR EQUATIONS
Some Exercises Explained [31 min.]
An Application: Writing Proper Rational Functions As Sums of Simpler Proper Rational Functions (Partial Fractions) [43 min.]

Lecture 34: SYSTEMS OF NON-LINEAR EQUATIONS
Solving (Mostly Graphically) a System of 2 Non-Linear Equations in 2 Variables [14 min.]

UNIT 7 - SOME DISCRETE TOPICS

Lecture 35: SEQUENCES
Infinite Sequences: Functions With Domain N [28 min.]
The Factorial Symbol: ! [8 min.]
Adding The First n Terms of a Sequence: nth Partial Sums & Summation Notation [17 min.]

Lecture 36: SEQUENCES
Arithmetic Sequences: Adding Your Way to Infinity [18 min.]
Geometric Sequences: Multiplying Your Way to Infinity [30 min.]

Lecture 37: SERIES & INDUCTION
Geometric Series & Their (Infinite) Sums [26 min.]
The Principle of Mathematical Induction [30 min.]

Lecture 38: THE BINOMIAL THEOREM
The “Binomial Coefficient” Symbol [11 min.]
Pascal’s Triangle [19 min.]
The Binomial Theorem: How to Expand (x+a)ⁿ [32 min.]

THE X-Y FILES: The Proof is in Here

Lecture 39 FOR UNIT 0 [29 min.]

• A Proof That Squareroot 2 Is Irrational [6 min.]
• A Proof That There Are The Same Number of Rational Numbers as Natural Numbers! [9 min.]
• A Proof That There Are More Real Numbers Than Natural Numbers! [14 min.]

FOR UNIT 2 [17 min.]

• Algebra For Science: Variation [17 min.]

FOR UNIT 4 [21 min.]

• How Can You Find Rational Zeros (if any) of a Polynomial Function? [21 min.]

FOR UNIT 7 [19 min.]

• A Proof That “e” Is Irrational [19 min.]

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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