INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith

Home » Mathematics » Algebra 1 » Multiplying Rational Expressions

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Multiplying Rational Expressions

  • To multiply two rational expressions, first factor all the numerators and denominators. Then cancel the common factors. Combine the remaining factors to form the final, simplified answer.

  • If two factors in the numerator and denominator look almost the same, factor –1 out of either of the factors and see if you get two identical factors.

Multiplying Rational Expressions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Procedure 0:16
    • Example
  • Cancel Before Multiplication 3:44
    • Example
  • Rational Expressions Containing Polynomials 6:19
    • Example
  • Lecture Example 1 11:09
  • Lecture Example 2 15:39
  • Additional Example 3
  • Additional Example 4

Related Books

Professor Fraser

Professor Fraser

Multiplying Rational Expressions

Slide Duration:

Table of Contents

Section 1: Basic Concepts
Variables and Expressions

24m 5s

Intro
0:00
Definitions
0:20
Variable
0:27
Algebraic Expression
2:21
Arithmetic Operations
2:27
Lecture Example 1
7:16
Lecture Example 2
8:27
Additional Example 3
-1
Additional Example 4
-2
Order of Operations

22m 16s

Intro
0:00
Why We Need It
0:15
Example: Order of Operations
1:25
Procedure To Evaluate
3:22
Lecture Example 1
6:45
Lecture Example 2
10:55
Additional Example 3
-1
Additional Example 4
-2
The Distributive Property

16m 27s

Intro
0:00
The Distributive Property
0:20
Extension to Several Numbers
2:05
Lecture Example 1
2:50
Lecture Example 2
5:15
Additional Example 3
-1
Additional Example 4
-2
The Real Number System

22m 35s

Intro
0:00
The Real Number System
0:21
Natural Numbers
0:22
Whole Numbers
0:37
Integers
1:06
Rational Numbers
1:35
Example: Decimals
2:07
The Real Number System, Cont.
5:00
Square Roots
5:10
Principle Square Root
6:04
Irrational Numbers
6:35
The Real Number System, Cont.
9:11
Picture Representation
9:25
Lecture Example 1
11:54
Lecture Example 2
14:10
Additional Example 3
-1
Additional Example 4
-2
Functions and Graphs

27m 49s

Intro
0:00
Functions
0:30
Definition
0:30
Example: Square Roots
1:18
Example: Gas Prices
3:20
Graphs
4:03
Definition
4:03
Example: Square Roots
4:48
Domain and Range
9:30
Definition
9:30
Example: Square Roots
10:01
Lecture Example 1
11:22
Lecture Example 2
13:58
Additional Example 3
-1
Additional Example 4
-2
Section 2: Solving Linear Equations
From Sentences to Equations

22m 13s

Intro
0:00
Strategy
1:40
Using Variables
1:55
Translating Phrases
2:18
Equality
3:00
Lecture Example 1
3:52
Lecture Example 2
8:36
Additional Example 3
-1
Additional Example 4
-2
Addition and Subtraction Techniques

14m 39s

Intro
0:00
Techniques
0:22
Adding the Same Number
0:28
Example
0:47
Subtracting the Same Number
1:50
Example
1:54
Strategy
2:36
Isolate the Variable
2:46
Lecture Example 1
3:48
Lecture Example 2
5:41
Additional Example 3
-1
Additional Example 4
-2
Multiplication and Division Techniques

19m 19s

Intro
0:00
Techniques
0:16
Multiplied by the Same Number
0:18
Example
0:26
Divided by the Same Number
1:23
Non-zero Exception
1:28
Dividing by Zero
2:17
Example
3:01
Strategy
3:51
Isolate the Variable
3:55
Lecture Example 1
4:55
Lecture Example 2
6:40
Additional Example 3
-1
Additional Example 4
-2
Techniques for Multi-Step Equations

19m 13s

Intro
0:00
What are Multi-step Equations?
0:08
Strategy
0:30
Opposite Operations
0:50
Last Operation
1:18
Lecture Example 1
1:54
Lecture Example 2
4:16
Additional Example 3
-1
Additional Example 4
-2
When the Variable is on Both Sides of the Equation

24m 16s

Intro
0:00
Solving More Complicated Equations
0:17
Distributive Property and Grouping Symbols
0:36
Combining
0:51
Moving Variables to One Side
1:06
Posibble Outcomes
1:50
Exactly One Solution
2:02
No Solution
2:35
Example
2:51
Identities-True for All Real Numbers
4:26
Example
4:45
Lecture Example 1
6:17
Lecture Example 2
9:39
Additional Example 3
-1
Additional Example 4
-2
Ratios and Proportions

22m 39s

Intro
0:00
Definitions
0:16
Ratio
0:21
Proportion
0:56
Example
1:13
Cross Products
3:00
Example: Original Proportion
3:12
Example 2
5:24
Rates
6:55
Example: Gas
7:22
Example 2: Apples
7:43
Example 3: Speed
8:16
Example 4: Cookies
8:46
Lecture Example 1
9:17
Lecture Example 2
11:53
Additional Example 3
-1
Additional Example 4
-2
Applications of Percents

20m 38s

Intro
0:00
Definitions
0:22
Percent of Change
0:23
Percent of Increase
0:42
Percent of Decrease
0:55
Examples of Percent of Change
1:14
Sales Tax
1:21
Discount
1:57
Lecture Example 1
2:51
Lecture Example 2
4:46
Additional Example 3
-1
Additional Example 4
-2
More Than One Variable

22m 34s

Intro
0:00
More Than One Variable
0:21
Strategy
1:02
Typical Application
1:33
Example: Circumference
1:48
Example: Perimeter
2:21
Lecture Example 1
2:53
Lecture Example 2
5:43
Additional Example 3
-1
Additional Example 4
-2
Section 3: Functions
Relations

24m 39s

Intro
0:00
Definition
0:17
List, Table, Graph
0:42
Example: List
0:56
Example: Table
2:00
Example: Graph
2:47
Domain and Range
4:10
Inverse of a Relation
5:01
Example
5:21
Lecture Example 1
7:26
Lecture Example 2
10:39
Additional Example 3
-1
Additional Example 4
-2
Functions

19m 3s

Intro
0:00
Definition
0:21
Function
0:24
Example: Relation
0:50
Vertical Line Test
1:48
Example: Pass
2:02
Example 2: Fail
3:32
Example 3: Fail
4:18
Function Notation
5:33
Proper Notation
5:43
Example
6:29
Domain and Range
7:32
Lecture Example 1
8:06
Lecture Example 2
9:39
Additional Example 3
-1
Additional Example 4
-2
Linear Functions

20m 3s

Intro
0:00
Definition
0:12
Linear Equation
0:15
Example: A and B not zero
0:27
Example: B is zero
0:58
Example: A is zero
1:13
Graph and Intercepts
1:43
Straight Line
1:52
x-intercepts, y-intercepts
2:19
Example: Intercepts
2:27
Graphing Linear Equations
4:19
Example
4:40
Linear Functions
7:23
Example
8:00
Lecture Example 1
10:11
Lecture Example 2
12:42
Additional Example 3
-1
Additional Example 4
-2
Section 4: Linear Functions and their Graphs
Slope and Rate of Change

23m 58s

Intro
0:00
Rate of Change
0:27
Independent Variable
0:37
Dependent Variable
0:51
Slope
2:02
Example
2:19
Steepness of the Line
4:56
Possible Slopes
5:58
Positive
6:13
Negative
6:34
Zero
6:50
Undefined
7:11
Lecture Example 1
8:03
Lecture Example 2
10:34
Additional Example 3
-1
Additional Example 4
-2
Direct Variation

13m 18s

Intro
0:00
Definitions
0:13
Direct Variation
0:14
Constant of Variation
0:26
Graph
1:02
Example: Positive and Negative k
1:18
Applications
1:49
Lecture Example 1
2:16
Lecture Example 2
4:39
Additional Example 3
-1
Additional Example 4
-2
Slope Intercept Form of an Equation

17m 11s

Intro
0:00
Slope Intercept Form
0:16
Example
0:37
Lecture Example 1
2:24
Lecture Example 2
5:43
Additional Example 3
-1
Additional Example 4
-2
Point Slope Form of an Equation

10m 32s

Intro
0:00
Point Slope Form
0:11
Example
0:50
Lecture Example 1
2:03
Lecture Example 2
3:03
Additional Example 3
-1
Additional Example 4
-2
Parallel and Perpendicular Lines

18m 15s

Intro
0:00
Parallel Lines
0:12
Example: Non-vertical Lines
0:32
Perpendicular lines
1:57
Example: Slope Product is -1
2:15
Negative Reciprocal
3:08
Lecture Example 1
3:58
Lecture Example 2
7:12
Additional Example 3
-1
Additional Example 4
-2
Section 5: Systems of Equations
Graphing Systems of Equations

20m 1s

Intro
0:00
System of Equations
0:09
Definition
0:15
Solution
0:59
Solving by Graphing
1:27
Example
2:33
Number of Solutions
3:37
Independent
3:58
Dependent
4:03
Inconsistent
4:08
Example: 1 Solution
4:15
Example: No Solution
4:34
Example: Infinite Solution
4:51
Lecture Example 1
5:39
Lecture Example 2
8:51
Additional Example 3
-1
Additional Example 4
-2
Solving by Substitution

24m 12s

Intro
0:00
Substitution
0:17
Example
0:51
Number of Solutions
1:43
Infinite Solutions
2:14
No Solutions
2:50
Lecture Example 1
3:40
Lecture Example 2
6:56
Additional Example 3
-1
Additional Example 4
-2
Solving by Addition and Subtraction

12m 18s

Intro
0:00
Fundamental Principle
0:18
Adding or Subtracting
0:50
Lecture Example 1
1:23
Lecture Example 2
4:03
Additional Example 3
-1
Additional Example 4
-2
Solving by Multiplication

19m 11s

Intro
0:00
Fundamental Principle
0:24
Multiplication
0:28
Lecture Example 1
1:40
Lecture Example 2
6:36
Additional Example 3
-1
Additional Example 4
-2
Applications: Systems of Equations

18m 24s

Intro
0:00
Solving Systems of Equations
0:08
Graphing
0:32
Addition
0:50
Subtraction
1:04
Multiplication
1:13
Substitution
1:30
Lecture Example 1
3:20
Lecture Example 2
4:27
Additional Example 3
-1
Additional Example 4
-2
Section 6: Inequalities
Inequalities: Addition and Subtraction Techniques

10m 8s

Intro
0:00
Fundamental Principle
0:11
Solutions of Inequalities
0:55
Inequality
1:02
Set Builder Notation
1:13
Example
1:15
Graph on a Number Line
2:13
Lecture Example 1
3:29
Lecture Example 2
5:14
Additional Example 3
-1
Additional Example 4
-2
Inequalities: Multiplication and Division Techniques

10m 59s

Intro
0:00
Fundamental Principle
0:11
Example: Positive Number
0:51
Fundamental Principle
1:35
Example: Negative Number
1:50
Lecture Example 1
4:05
Lecture Example 2
5:12
Additional Example 3
-1
Additional Example 4
-2
Techniques for Multi-Step Inequalities

16m 21s

Intro
0:00
Similarity to Multi-step Equations
0:15
Inequalities Containing Grouping Symbols
1:03
Special Cases
1:26
Solution: All Real Numbers
1:40
Solution: Empty Set
2:16
Lecture Example 1
2:54
Lecture Example 2
4:37
Additional Example 3
-1
Additional Example 4
-2
Compound Inequalities

23m

Intro
0:00
Inequalities Combined by 'And'
0:12
Three Ways to Write Solution Set
1:41
Set Notation
2:01
Inequalities
2:12
Graph
2:29
Examples
2:31
Inequalities Combined by 'Or'
4:45
Example
5:59
Lecture Example 1
7:40
Lecture Example 2
10:55
Additional Example 3
-1
Additional Example 4
-2
Equations with Absolute Values

23m 13s

Intro
0:00
Absolute Value
0:15
Example: Distance
1:00
Absolute Value Function
2:34
Example
3:05
Lecture Example 1
4:57
Lecture Example 2
8:28
Additional Example 3
-1
Additional Example 4
-2
Inequalities with Absolute Values

17m 34s

Intro
0:00
Inequalities of the Form |x|< n
0:22
Inequalities of the Form |x|> n
3:30
Lecture Example 1
5:33
Lecture Example 2
8:50
Additional Example 3
-1
Additional Example 4
-2
Graphing Inequalities with Two Variables

30m 50s

Intro
0:00
Graph
0:22
Half-Plane
0:28
Boundary
1:37
Technique for Graphing
3:14
Example
4:01
Dashed Line
5:13
Solid Line
6:30
Technique for Graphing, cont.
8:58
Example
9:09
Lecture Example 1
12:19
Lecture Example 2
15:48
Additional Example 3
-1
Additional Example 4
-2
Graphing Systems of Inequalities

27m 38s

Intro
0:00
System of Inequalities
0:26
Example
0:43
Solving a System of Inequalities
1:17
Example
1:41
Lecture Example 1
2:58
Lecture Example 2
7:22
Additional Example 3
-1
Additional Example 4
-2
Section 7: Polynomials
Multiplying Monomials

19m 55s

Intro
0:00
What is a Monomial?
0:43
Constant
1:11
Power
1:48
Exponent and Base
2:39
Properties of Exponents
3:02
Simplified Form
5:52
Each Base Appears Exactly Once
6:02
No Powers of Powers
6:18
All Fractions Simplified
6:33
Lecture Example 1
6:43
Lecture Example 2
8:12
Additional Example 3
-1
Additional Example 4
-2
Dividing Monomials

23m 9s

Intro
0:00
Properties of Exponents
0:26
Example
1:07
Properties of Exponents, cont
2:18
Example
3:18
Properties of Exponents, cont
4:55
Example
5:32
Properties of Exponents, cont
8:49
Lecture Example 1
11:43
Lecture Example 2
13:13
Additional Example 3
-1
Additional Example 4
-2
Polynomials

16m 35s

Intro
0:00
What is a Polynomial
0:45
Monomial
0:47
Example: Trinomial
1:46
Trinomial
2:10
Example: Binomial
2:21
Binomial
2:30
Degree of a Polynomial
4:28
Example
4:40
Ordering Polynomials
6:45
Example
7:27
Lecture Example 1
9:27
Lecture Example 2
10:31
Additional Example 3
-1
Additional Example 4
-2
Adding and Subtracting Polynomials

22m

Intro
0:00
Adding Polynomials
0:36
Like Polynomials
0:48
Example
1:04
Subtracting Polynomials
3:55
Example
4:01
Lecture Example 1
5:30
Lecture Example 2
8:04
Additional Example 3
-1
Additional Example 4
-2
Multiplying Polynomials by Monomials

24m 8s

Intro
0:00
Distributive Property
0:22
Basic Principle
0:42
Example
1:04
Solving Equations
3:25
Lecture Example 1
4:26
Lecture Example 2
7:51
Additional Example 3
-1
Additional Example 4
-2
Multiplying Polynomials

26m 37s

Intro
0:00
Distributive Property
0:23
Example
0:34
Example
1:55
Foil Method
5:17
Lecture Example 1
7:44
Lecture Example 2
10:15
Additional Example 3
-1
Additional Example 4
-2
Special Products

19m 40s

Intro
0:00
Square of a Sum
0:12
Perfect Square Trinomial
1:43
Example
2:13
FOIL Technique
3:09
Square of a Difference
3:38
Example
5:27
Difference of Two Squares
6:12
Example
7:43
Lecture Example 1
8:51
Lecture Example 2
11:06
Additional Example 3
-1
Additional Example 4
-2
Section 8: Factoring
Factoring Monomials

27m 25s

Intro
0:00
Prime and Composite Numbers
1:01
Prime Number
1:23
Composite Number
3:03
Factored Forms
4:07
Prime Factored Form
4:19
Factored Form
5:40
Greatest Common Factor (GCF)
7:41
Example
8:00
GCF of Monomials
10:07
Example
10:18
Lecture Example 1
12:52
Lecture Example 2
17:08
Additional Example 3
-1
Additional Example 4
-2
Factoring Using Greatest Common Factor (GCF)

26m 3s

Intro
0:00
Distributive Property
0:19
Factor Out an Expression
1:21
Example
2:52
Factoring by Grouping
4:40
Example
6:00
Zero Product Property
8:24
Example
9:03
Lecture Example 1
11:19
Lecture Example 2
14:19
Additional Example 3
-1
Additional Example 4
-2
Factoring Trinomials with Leading Coefficient 1

27m 37s

Intro
0:00
Factoring Trinomials
0:23
Example
2:06
Rules for Signs
6:27
Both Positive
6:41
Both Negative
7:00
Opposite Signs
7:33
Solving Equations
8:26
Example
8:59
Lecture Example 1
11:14
Lecture Example 2
14:01
Additional Example 3
-1
Additional Example 4
-2
Factoring General Trinomials

34m 11s

Intro
0:00
Factoring Trinomials
0:40
Example: List
1:38
Grouping
5:46
Example
5:55
Rules for Signs
9:04
Example
9:22
Greatest Common Factor (GCF)
10:29
Prime Polynomials
11:03
Example
11:32
Solving Equations
12:32
Lecture Example 1
13:11
Lecture Example 2
18:37
Additional Example 3
-1
Additional Example 4
-2
Factoring the Difference of Two Squares

20m 22s

Intro
0:00
Difference of Two Squares
0:40
Example
1:56
Factoring Using Several Techniques
3:33
Solving Equations
4:11
Lecture Example 1
4:55
Lecture Example 2
6:05
Additional Example 3
-1
Additional Example 4
-2
Factoring Perfect Squares

18m 7s

Intro
0:00
Perfect Squares
0:10
Perfect Square Trinomials
1:17
Solving Equations
2:53
Square Root Property
2:58
Example
3:06
Lecture Example 1
4:23
Lecture Example 2
6:56
Additional Example 3
-1
Additional Example 4
-2
Section 9: Quadratic Functions
Graphing Quadratic Functions

28m 5s

Intro
0:00
Parabolas
0:09
Parabolas That Open Upward
3:58
Minimum
4:26
Parabolas That Open Downward
4:54
Maximum
5:09
Vertex
7:28
Example
8:01
Axis of Symmetry
11:03
Lecture Example 1
13:27
Lecture Example 2
17:17
Additional Example 3
-1
Additional Example 4
-2
Solving Equations by Graphing

28m 18s

Intro
0:00
Solving a Quadratic Equation
0:22
Example
0:50
Two Distinct Solutions/Roots
5:08
One Double Root
5:58
No Real Roots
7:14
Estimating Solutions
8:00
Example
8:16
Example
8:31
Lecture Example 1
9:43
Lecture Example 2
14:10
Additional Example 3
-1
Additional Example 4
-2
Solving Equations by Completing the Square

25m 7s

Intro
0:00
Perfect Square Trinomials
0:22
Example
1:12
Completing the Square
3:17
Example
3:25
Solving Equations
6:12
Example
6:27
Leading Coefficient is 1
8:25
Lecture Example 1
10:23
Lecture Example 2
14:17
Additional Example 3
-1
Additional Example 4
-2
Solving Equations Using the Quadratic Formula

24m 15s

Intro
0:00
Quadratic Formula
1:26
Example
2:23
Discriminant
4:28
Two Distinct Real Roots
5:07
One Double Real Root
5:23
No Real Root
5:42
Why Does It Work?
6:05
Lecture Example 1
7:53
Lecture Example 2
11:50
Additional Example 3
-1
Additional Example 4
-2
Section 10: Radical Expressions and Equations
Simplifying Radical Expressions

32m 40s

Intro
0:00
Radical Expressions
0:14
Radicand
0:28
Example
0:32
Simplest Form
1:07
Example
1:35
Product Property
2:34
Verifications
2:47
Square Roots of Variables with Even Powers
4:27
Quotient Rule
6:45
Example
6:55
Rationalizing Denominators
7:32
Example
7:41
Conjugates
10:44
Examples
10:59
Simplest Radical Form
14:44
Example
15:39
Lecture Example 1
17:34
Lecture Example 2
19:38
Additional Example 3
-1
Additional Example 4
-2
Operations with Radical Expressions

19m 59s

Intro
0:00
Adding and Subtracting Radical Expressions
0:12
Example
0:19
Multiplying Radical Expressions
1:53
Example
2:07
Lecture Example 1
5:29
Lecture Example 2
6:36
Additional Example 3
-1
Additional Example 4
-2
Solving Radical Equations

26m 50s

Intro
0:00
Radical Equations
0:22
Example
0:33
Solving a Radical Equation
2:04
Example
2:16
Extraneous Solutions
3:14
Example
4:31
Lecture Example 1
7:33
Lecture Example 2
9:45
Additional Example 3
-1
Additional Example 4
-2
Pythagorean Theorem

19m 8s

Intro
0:00
Right Triangles
0:51
Right Angle
1:03
Vertex
1:11
Symbol
1:16
Hypotenuse and Legs
1:42
Pythagorean Theorem
2:27
Example
2:47
Example
3:12
Pythagorean Triples
4:03
Converse of the Pythagorean Theorem
6:39
Example
7:41
Lecture Example 1
8:57
Lecture Example 2
10:05
Additional Example 3
-1
Additional Example 4
-2
Distance Formula

18m 27s

Intro
0:00
Distance Formula
0:17
Missing Coordinates
1:46
Example
1:55
Lecture Example 1
4:10
Lecture Example 2
5:57
Additional Example 3
-1
Additional Example 4
-2
Section 11: Rational Expressions and Equations
Inverse Variation

14m 43s

Intro
0:00
Inverse Variation
0:23
Constant of Variation
0:37
Graphing Inverse Variation
1:08
Example: Hyperbola
1:15
Product Rule
3:54
Lecture Example 1
5:31
Lecture Example 2
7:43
Additional Example 3
-1
Additional Example 4
-2
Rational Expressions

32m 43s

Intro
0:00
Rational Expression
0:55
Example
1:03
Excluded Values
1:29
Examples
1:40
Simplifying Rational Expressions
5:36
Greatest Common Factor (GCF)
5:46
Example
5:55
Example
7:42
Simplifying and Excluded Values
9:34
Lecture Example 1
9:50
Lecture Example 2
13:55
Additional Example 3
-1
Additional Example 4
-2
Multiplying Rational Expressions

27m 22s

Intro
0:00
Procedure
0:16
Example
0:33
Cancel Before Multiplication
3:44
Example
3:55
Rational Expressions Containing Polynomials
6:19
Example
6:26
Lecture Example 1
11:09
Lecture Example 2
15:39
Additional Example 3
-1
Additional Example 4
-2
Dividing Rational Expressions

28m 30s

Intro
0:00
Procedure
0:21
Example
0:59
Cancel Before Multiplication
4:31
Example
4:39
Rational Expressions Containing Polynomials
8:01
Example
8:08
Lecture Example 1
10:43
Lecture Example 2
13:46
Additional Example 3
-1
Additional Example 4
-2
Dividing Polynomials

29m 44s

Intro
0:00
Dividing a Polynomial by a Monomial
1:07
Example
1:14
Dividing a Polynomial by a Binomial
3:30
Example
3:51
Long Division
5:20
Example
6:25
Missing Terms
11:11
Example
12:25
Lecture Example 1
14:54
Lecture Example 2
16:35
Additional Example 3
-1
Additional Example 4
-2
Adding and Subtracting Rational Expressions with Like Denominators

16m 50s

Intro
0:00
Adding with Like Denominators
0:14
Example
0:41
Subtraction with Like Denominators
2:23
Example
2:30
Denominators That Are Additive Inverses
2:55
Example
3:03
Lecture Example 1
6:41
Lecture Example 2
7:09
Additional Example 3
-1
Additional Example 4
-2
Adding and Subtracting Rational Expressions with Unlike Denominators

36m

Intro
0:00
Least Common Multiple (LCM) of Polynomials
0:34
Example
0:45
Example
1:55
Adding and Subtracting
4:08
Example
4:45
Lecture Example 1
8:13
Lecture Example 2
10:22
Additional Example 3
-1
Additional Example 4
-2
Complex Fractions

18m 43s

Intro
0:00
Mixed Expression
0:08
Example
0:30
Complex Fraction
2:45
Example
2:57
Example
3:11
Example
3:22
Simplifying Complex Fractions
3:58
Why Does It Work?
4:22
Lecture Example 1
4:48
Lecture Example 2
8:44
Additional Example 3
-1
Additional Example 4
-2
Rational Equations

39m 10s

Intro
0:00
Definition
0:08
Examples
0:17
Solving Rational Expressions
3:02
Example
3:09
Work Problems
6:31
Example
6:48
Extraneous Solutions
13:12
Example
14:13
Lecture Example 1
17:21
Lecture Example 2
19:59
Additional Example 3
-1
Additional Example 4
-2
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