College Algebra

A series of free College Algebra Video Lessons from UMKC - The University of Missouri-Kansas City.

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 EXPRESSIONS 

• 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 Squareoot 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.]

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