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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Multiplying and Dividing Rational Expressions

Slide Duration:

Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
II. Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
III. Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
IV. Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
V. Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
VI. Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
VII. Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
VIII. Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
IX. Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
X. Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
XI. Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (9)

1 answer

Last reply by: Dr Carleen Eaton
Sun Jul 8, 2018 12:59 PM

Post by John Stedge on June 12 at 02:00:12 PM

In example one can't you simplify further and make the answer -4/3 instead of -x+4/x+3?

1 answer

Last reply by: Dr Carleen Eaton
Sun Feb 11, 2018 9:50 PM

Post by Shiden Yemane on February 10 at 12:32:39 PM

Thanks Dr. Eaton! I've learned a lot from your Algebra 2 series.

0 answers

Post by julius mogyorossy on December 25, 2014

Merry Christmas Dr. Carleen, thanks for everything, I can't wait till you know who I am, how awesome I am. I got my math notes on my iPod, now if I can only find where apple is hiding them from me, apple is always hiding things from me. I can't wait to take the Clep test. I wish I could hire you as my coach.

1 answer

Last reply by: Dr Carleen Eaton
Thu Mar 27, 2014 6:46 PM

Post by Taslim Yakub on February 21, 2014

why does the 4+x become x+4 and not have a negative sign in front of it.

1 answer

Last reply by: Dr Carleen Eaton
Wed Jan 1, 2014 1:00 AM

Post by Myriam Bouhenguel on December 27, 2013

for example 3 it is supposed to be (2x+1) (x-3) for the factorization of 2x^2+5x-3 at 34:37

Multiplying and Dividing Rational Expressions

  • To simplify an algebraic fraction, factor the numerator and denominator completely. Then cancel common factors. When multiplying or dividing, do the same thing – factor all the numerators and denominators, then cancel common factors.
  • 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 and Dividing Rational Expressions

Simplify [(n2 − 10n + 21)/(n − 3)]
  • Factor the numerator
  • [(n2 − 10n + 21)/(n − 3)] = [((n − )(n − ))/(n − 3)]
  • [(n2 − 10n + 21)/(n − 3)] = [((n − 3)(n − 7))/(n − 3)]
  • Cancel out common factors
  • [((n − 3)(n − 7))/(n − 3)] = [((n − 7))/] 
n − 7
Simplify [(a + 3)/(a2 − 9)]
  • Factor the denominator
  • [(a + 3)/(a2 − 9)] = [(a + 3)/((a + )(a − ))]
  • [(a + 3)/(a2 − 9)] = [(a + 3)/((a + 3)(a − 3))]
  • Cancel out common factors
  • [(a + 3)/((a + 3)(a − 3))] = [/((a − 3))] 
[1/(a − 3)]
Simplify [(p + 5)/(5p2 − 25p)] ×[(p2 − 7p + 10)/(5p + 25)]
  • Factor the numerator and denominator completely
  • [(p + 5)/(5p2 − 25p)] ×[(p2 − 7p + 10)/(5p + 25)] = [(p + 5)/(5p( − ))] ×[((p − )(p − ))/(5(p + ))]
  • [(p + 5)/(5p2 − 25p)] ×[(p2 − 7p + 10)/(5p + 25)] = [(p + 5)/(5p(p − 5))] ×[((p − 2)(p − 5))/(5(p + 5))]
  • Multiply
  • [(p + 5)/(5p(p − 5))] ×[((p − 2)(p − 5))/(5(p + 5))] = [((p + 5)(p − 2)(p − 5))/(5p(p − 5)5(p + 5))]
  • Cancel out common factors
  • [((p + 5)(p − 2)(p − 5))/(5p(p − 5)5(p + 5))] = [((p − 2))/5p5] 
  • Simplify
[(p − 2)/25p]
Simplify [(3x + 12)/(3x + 9)] ×[(2x3 − 2x2)/(2x3 + 8x2)]
  • Factor the numerator and denominator completely
  • [(3x + 12)/(3x + 9)] ×[(2x3 − 2x2)/(2x3 + 8x2)] = [(3( + ))/(3( + ))] ×[(2x2( − ))/(2x2( + ))]
  • [(3x + 12)/(3x + 9)] ×[(2x3 − 2x2)/(2x3 + 8x2)] = [(3(x + 4))/(3(x + 3))] ×[(2x2(x − 1))/(2x2(x + 4))]
  • Multiply
  • [(3(x + 4))/(3(x + 3))] ×[(2x2(x − 1))/(2x2(x + 4))] = [(3(x + 4)2x2(x − 1))/(3(x + 3)2x2(x + 4))]
  • Cancel out common factors
  • [(3(x + 4)2x2(x − 1))/(3(x + 3)2x2(x + 4))] = [((x − 1))/((x + 3))] 
  • Simplify
[(x − 1)/(x + 3)]
Simplify [(x2 − 6x + 8)/(x2 − 16)] ×[(x + 4)/(5x2 + 25x)]
  • Factor the numerator and denominator completely
  • [(x2 − 6x + 8)/(x2 − 16)] ×[(x + 4)/(5x2 + 25x)] = [((x − )(x − ))/(( + )( − ))] ×[(x + 4)/(5x( + ))]
  • [(x2 − 6x + 8)/(x2 − 16)] ×[(x + 4)/(5x2 + 25x)] = [((x − 4)(x − 2))/((x + 4 )(x − 4))] ×[(x + 4)/(5x(x + 5))]
  • Multiply
  • [((x − 4)(x − 2)( x + 4 ))/((x + 4 )(x − 4)5x(x + 5))]
  • Cancel out common factors
  • [((x − 4)(x − 2)( x + 4 ))/((x + 4 )(x − 4)5x(x + 5))] = [((x − 2))/(5x(x + 5))] 
  • Simplify
[(x − 2)/(5x(x + 5))]
Simplify [(x2 − 1)/(x2 + 2x + 1)] ×[(x2 + 2x + 1)/(x + 1)]
  • Factor the numerator and denominator completely
  • [(x2 − 1)/(x2 + 2x + 1)] ×[(x2 + 2x + 1)/(x + 1)] = [((x + )(x − ))/(( + )( + ))] ×[(( + )( + ))/(x + 1)]
  • [(x2 − 1)/(x2 + 2x + 1)] ×[(x2 + 2x + 1)/(x + 1)] = [((x + 1)(x − 1))/((x + 1)(x + 1))] ×[((x + 1)(x + 1))/(x + 1)]
  • Multiply
  • [((x + 1)(x − 1))/((x + 1)(x + 1))] ×[((x + 1)(x + 1))/(x + 1)] = [((x + 1)(x − 1)(x + 1)(x + 1))/((x + 1)(x + 1)( x + 1 ))]
  • Cancel out common factors
  • [((x + 1)(x − 1)(x + 1)(x + 1))/((x + 1)(x + 1)( x + 1 ))] = [((x − 1))/]
  • Simplify
x − 1
Simplify [(3x2 − 9x)/(x2 − 7x + 12)] ÷[(5x2 − 20x)/(x − 4)]
  • Factor the numerator and denominator completely
  • [(3x2 − 9x)/(x2 − 7x + 12)] ÷[(5x2 − 20x)/(x − 4)] = [(3x( − ))/(( − )( − ))] ÷[(5x( − ))/(x − 4)]
  • [(3x2 − 9x)/(x2 − 7x + 12)] ÷[(5x2 − 20x)/(x − 4)] = [(3x(x − 3))/((x − 3)(x − 4))] ÷[(5x(x − 4))/(x − 4)]
  • Get rid of the division by multiplying by the recriprocal of the second rational
  • [(3x(x − 3))/((x − 3)(x − 4))] ÷[(5x(x − 4))/(x − 4)] = [(3x(x − 3))/((x − 3)(x − 4))] •[((x − 4))/(5x(x − 4))] = [(3x(x − 3)(x − 4))/((x − 3)(x − 4)5x(x − 4))]
  • Cancel out common factors
  • [(3x(x − 3)(x − 4))/((x − 3)(x − 4)5x(x − 4))] = [3/((x − 4)5)]
  • Simplify
[3/(5(x − 4))]
Simplify [(x2 + x − 12)/(x − 3)] ÷[(x2 + 6x + 8)/(x2 + 4x + 4)]
  • Factor the numerator and denominator completely
  • [(x2 + x − 12)/(x − 3)] ÷[(x2 + 6x + 8)/(x2 + 4x + 4)] = [(( + )( − ))/(x − 3)] ÷[(( + )( + ))/(( + )( + ))]
  • [(x2 + x − 12)/(x − 3)] ÷[(x2 + 6x + 8)/(x2 + 4x + 4)] = [((x + 4)(x − 3))/(x − 3)] ÷[((x + 2)(x + 4))/((x + 2)(x + 2))]
  • Get rid of the division by multiplying by the recriprocal of the second rational
  • [((x + 4)(x − 3))/(x − 3)] ÷[((x + 2)(x + 4))/((x + 2)(x + 2))] = [((x + 4)(x − 3))/(x − 3)] = [((x + 4)(x − 3)(x + 2)(x + 2))/(( x − 3 )(x + 2)(x + 4))]
  • Cancel out common factors
  • [((x + 4)(x − 3)(x + 2)(x + 2))/(( x − 3 )(x + 2)(x + 4))] = [((x + 2))/]
  • Simplify
x + 2
Simplify [([3/(3x + 15)])/([(3x + 6)/(2x + 10)])]
  • Rewrite the complex fraction by dividing the numerator by the numerator using the division sign ÷
  • [([3/(3x + 15)])/([(3x + 6)/(2x + 10)])] = [3/(3x + 15)] ÷[(3x + 6)/(2x + 10)]
  • Factor the numerator and denominator completely
  • [3/(3x + 15)] ÷[(3x + 6)/(2x + 10)] = [3/(3( + ))] ÷[(3( + ))/(2( + ))]
  • [3/(3x + 15)] ÷[(3x + 6)/(2x + 10)] = [3/(3(x + 5))] ÷[(3(x + 2))/(2(x + 5))]
  • Get rid of the division by multiplying by the recriprocal of the second rational
  • [3/(3(x + 5))] ÷[(3(x + 2))/(2(x + 5))] = [3/(3(x + 5))] •[(2(x + 5))/(3(x + 2))] = [(3*2(x + 5))/(3(x + 5)*3(x + 2))]
  • Cancel out common factors
  • [(3*2(x + 5))/(3(x + 5)*3(x + 2))] = [*2/(*3(x + 2))]
  • Simplify
[2/(3(x + 2))]
Simplify [([5x/(25x2 + 20x)])/([(5x − 15)/(25x + 20)])]
  • Rewrite the complex fraction by dividing the numerator by the numerator using the division sign ÷
  • [([5x/(25x2 + 20x)])/([(5x − 15)/(25x + 20)])] = [5x/(25x2 + 20x)] ÷[(5x − 15)/(25x + 20)]
  • Factor the numerator and denominator completely
  • [5x/(25x2 + 20x)] ÷[(5x − 15)/(25x + 20)] = [5x/(5x( + ))] ÷[(5( − ))/(5( + ))]
  • [5x/(25x2 + 20x)] ÷[(5x − 15)/(25x + 20)] = [5x/(5x(5x + 4))] ÷[(5(x − 3))/(5(5x + 4))]
  • Get rid of the division by multiplying by the recriprocal of the second rational
  • [5x/(5x(5x + 4))] ÷[(5(x − 3))/(5(5x + 4))] = [5x/(5x(5x + 4))] •[(5(5x + 4))/(5(x − 3))] = [(5x*5(5x + 4))/(5x*(5x + 4)5(x − 3))]
  • Cancel out common factors
  • [(5x*5(5x + 4))/(5x*(5x + 4)5(x − 3))] = [*/(*(x − 3))]
  • Simplify
[1/(x − 3)]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Multiplying and Dividing 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
  • Simplifying Rational Expressions 0:22
    • Algebraic Fraction
    • Examples: Rational Expressions
    • Example: GCF
    • Example: Simplify Rational Expression
  • Factoring -1 4:04
    • Example: Simplify with -1
  • Multiplying and Dividing Rational Expressions 6:59
    • Multiplying and Dividing
    • Example: Multiplying Rational Expressions
    • Example: Dividing Rational Expressions
  • Factoring 14:01
    • Factoring Polynomials
    • Example: Factoring
  • Complex Fractions 18:22
    • Example: Numbers
    • Example: Algebraic Complex Fractions
  • Example 1: Simplify Rational Expression 25:56
  • Example 2: Simplify Rational Expression 29:34
  • Example 3: Simplify Rational Expression 31:39
  • Example 4: Simplify Rational Expression 37:50

Transcription: Multiplying and Dividing Rational Expressions

Welcome to Educator.com.0000

Today, we are going to start a series of lectures on rational expressions and equations, starting out with multiplying and dividing rational expressions.0002

And this may be review from a topic you have had before; but we are going to go ahead and go over it,0011

and then go on to some more advanced topics with rational expressions.0017

I am beginning with simplifying rational expressions, and the definition, first, of a rational expression.0022

A rational expression is actually the ratio of two polynomial expressions.0030

So, it is a quotient; in this case, it is a quotient whose numerator and denominator are polynomials.0035

So, you can just look at this as an algebraic fraction.0042

Consider an example such as 4x divided by (3x - 5): this is a rational expression.0048

The numerator is a polynomial, and the denominator is a polynomial, and it is an algebraic fraction.0057

In algebra, we can think of polynomials as the integers, and then rational expressions as fractions.0064

Another example might be something like (2x2 - x + 9), over (x2 - 7).0072

Now, recall that, when a fraction is simplified, it has no common factor between the numerator and the denominator, other than 1 or -1.0084

So, let's start out by reviewing how this would work with numbers, and then applying these same concepts to rational expressions.0094

If I look at a fraction such as 10/15, you will immediately recognize that this is not in simplest form.0101

And you might even simplify it without really thinking about the exact steps you are taking.0110

Let's go through those steps, so we can apply them here.0115

If I go ahead and factor this out, I will end up with (5 times 2) over (5 times 3).0119

And what I am going to end up doing is canceling out that common factor to end up with 2/3.0128

And this is in simplest form, because now these numbers, 2 and 3, no longer have any common factor, other than 1 or -1.0136

So, it is the same idea here: to simplify a rational expression, you are going to divide the numerator and denominator by their greatest common factor.0146

Let's look at an example, now, involving a rational expression.0154

The first step is to factor: and the numerator here is already factored, and you will recall that0161

x2 - 9 is going to factor into (x + 3) (x - 3).0173

And as I look, I see that I do have a common factor; I have a common binomial factor of (x + 3).0185

So, the same way that I canceled out the common factor here, I am going to do the same here.0192

Now, remember that, when I cancel this out, what I am going to end up with is actually 1 in the numerator,0199

because, although we don't write it out, there actually is a 1 here (1 times x + 3).0205

So, this isn't just going to become 0 in the numerator; once I cancel the (x + 3)s out, I am going to end up with 1/(x - 3).0212

Now, this rational expression is in simplest form.0220

I also want you to notice that, even though there are no variables in the numerator, this is still a rational expression.0224

It is still an algebraic fraction; we end up with the rational expression that we started out with, now in simplest form: 1 divided by (x - 3).0230

Sometimes, factoring out a -1 will help to simplify a rational expression.0245

You might look at a rational expression and think, "Well, there are no common factors."0249

But there are some factors that look pretty close, except the signs are off.0254

So, let's look at an example: if I start out with the rational expression (2 - x)/(3x - 6), and I am asked to simplify this,0260

my first step is always going to be to factor; the numerator is already factored;0277

in the denominator, I have a common factor of 3, so I can factor out a 3.0283

That leaves behind an x, and here a -2.0289

Now, to look at this a little bit more easily, and compare what I have in the numerator and the denominator,0294

I am going to start out by rewriting this with the x first.0299

So, it is actually going to be -x + 2; this allows me to more easily compare this with what I have in the denominator, which is 3 times (x - 2).0304

When I compare this binomial factor in the numerator and the denominator, I see that the only difference here0321

is that the x's are opposite signs (so look at it like this: -x and x), and the 2's are also opposite signs.0328

Therefore, if I factor out a -1 from either the numerator or the denominator (one or the other--I am going to pick the numerator),0338

I can end up with what is inside the parentheses--that factor--having the same sign.0349

So, instead of this being 1 times this, I am going to factor out this -1, and I will get -1.0354

So, if I pull the -1 out, I am going to end up with an (x - 2).0362

And I see that I did this correctly, because if I multiply -1 times x, I am going to get a -x back.0366

If I multiply -1 times -2, I will get a + 2 back.0372

I chose to do the numerator; I could have left the numerator alone and factored the -1 out of the denominator, which would have yielded -x + 2.0376

So, here I factored this from the numerator; and this gives me, now, a common factor, (x - 2)/(x - 2).0386

I can then cancel these out; and this simplifies to -1/3.0395

Here, I looked, and I saw that I had what would have been the same factor, except that the signs on the terms were opposite.0401

In that case, factoring out a -1 will allow you to simplify the rational expression that you are working with.0410

Simplifying is tied closely to working with rational expressions in terms of multiplication and division.0420

Let's start out by reviewing the rules that are involved with multiplying and dividing rational expressions,0430

and also thinking back to how it works when you are multiplying fractions that involve numbers.0439

A lot of the concepts are the same.0445

So, if a/b and c/d are rational expressions, then in order to multiply these two (in order to multiply two rational expressions),0448

what you are going to do is multiply the numerators and multiply the denominators, and end up with something in the form ac/bd.0458

Notice that there is the restriction that b cannot equal 0 and d cannot equal 0,0473

because, as usual, we cannot allow the denominator to become 0, because that would be undefined.0478

Let's first talk just about the multiplication; and then we can go on and talk about division.0485

OK, so first, we are just going to talk about multiplying or dividing rational expression that just have terms that are monomials in the numerator or denominator.0495

And then, we will go on to talk about when you actually have a polynomial that is a binomial, trinomial, or greater in the numerator and/or denominator.0503

Let's start out with this example: 4xy3 times x2.0514

And that is going to give me...times 12xy5, divided by x4.0526

In order to multiply this, what I am going to do is actually, first, cancel out any common factors that I might have.0539

So, to make this easier, instead of trying to multiply x2 times x4, I am just going to go ahead and cancel out common factors.0552

So, I see here that I have an x here, and I have an x2 here; they have a common factor of x.0562

Here, I have another x, and this x cancels.0568

Then, I am just going to multiply what is left behind.0572

And this is going to leave me with 4y3 times 12y5, divided by x4.0577

Now, I multiply what I have left, which is 4 times 12, and that is going to give me 48.0591

And then, recall that multiplying exponential expressions that have the same base0600

means that I am going to add the exponents, so this is going to give me y8 divided by x4.0606

So, to multiply, you should first cancel out common factors.0613

Then, go on and multiply what is left in the numerator and what is left in the denominator.0618

Looking at division: if I have rational expressions a/b, divided by c/d, I can handle this by turning this into a multiplication problem.0624

And that is the same way that we handle division...0639

Let's say we were asked to divide 3/4 divided by 5/8.0641

I would handle that by rewriting it as 3/4 times the inverse, the reciprocal, of 5/8, which is 8/5.0649

Once I get to there, I just follow my usual rules for multiplication.0659

where I am going to cancel out common factors, and then that becomes the numerator, 3 times 2, divided by the denominator, 5 times 1.0663

This gives me 6/5.0673

I am going to do the same thing here, only I would be working with rational expressions.0675

So, if I have something like 2xy2 divided by 4x2y3 times xy divided by 8x2,0680

divided by...if I am being asked to divide these, I am simply going to rewrite this0703

as 2xy2 divided by 4x2y3; so I keep that first rational expression the same.0708

And then, I multiply it by its inverse, which is 8x2 divided by xy.0716

From there, I am going to proceed, just as I did up there, with multiplication.0725

And I am going to cancel out common factors and then multiply the numerators and denominators of what remains behind.0731

OK, so I have here a common factor of 4; that cancels to 1; this becomes 2.0740

I also see that I have an x2 term in the numerator and the denominator; those cancel.0748

I have an x in the numerator and an x in the denominator; those cancel.0755

And then, I have a y in the numerator and in the denominator; those cancel.0760

I have made my life a lot simpler, because instead of multiplying higher powers and larger constants, I have gotten rid of a lot of that.0767

So, now it comes time to multiply what is left behind.0775

Here I have a 2 and a y; all I have left over here is actually the constant 2; in the denominator, I have a 1--0778

I don't need to write that: I can just write the y3.0785

And actually, I have one more common factor left: so 4y/y3.0789

And if you see that, then you can go ahead and simplify at the end, since this common factor wasn't already canceled.0796

Then I am going to go ahead and simplify even further to give me 4 divided by y2.0802

So again, division just involves keeping the first rational expression the same, taking the inverse of the second rational expression...0810

and then I went ahead and canceled out the common factors I found,0823

multiplied the numerator and the denominator, and then checked to make sure that this was simplified.0826

And then, I found another common factor, which I went ahead and took care of there.0831

Factoring: if the rational expressions contain polynomials, you actually may need to factor them before a product or quotient can be simplified.0841

So, in the last problem, I showed how I canceled out common factors before I went ahead0850

and did the multiplication of the numerator and the multiplication of the denominator.0855

I want to do the same thing if I am handling polynomials.0859

Only, in this case, in order to do that canceling out, in order to even find what my factors are,0863

I am going to need to probably factor both the numerator and the denominator, if they are not already factored.0868

For example, 2x2 - 6x - 20, divided by x2 - 49:0876

if I am asked to multiply that by, say, 5x - 35, divided by x2 + 4x + 5,0888

last time we were working with monomials, and then I just went ahead and started canceling out my factors;0899

here, I don't even know what my factors are yet.0904

So, I am going to write these in factored form.0906

I can pull the common factor of 2 out from this trinomial to get x2 - 3x - 10.0910

x2 - 49 I am going to recognize as (x + 7) times (x - 7).0920

5x - 21 I can factor out a 5--that gives me (x - 7) times 5.0929

Here, I actually have...let's see...this is x2 + 4x + 5; let's actually make that a 4 right here...(x + 2) times (x + 2).0935

Let's go ahead and make that simpler to work with.0954

Now, I am looking, and I am seeing that my factoring is not done; I can still factor a little bit more up here.0958

This is going to be an x and an x; and this is negative, so I am going to have a positive here and a negative here.0965

And what I am going to look at is the factors of 10: 1 and 10, and 2 and 5.0972

And when one is positive and one is negative, what combination will give me -3?0977

Well, I can see that 1 and 10 are too far apart, so I am going to work with 2 and 5.0982

And I want to end up with a negative number, so I will make the 5, the larger number, negative.0987

And -5 plus 2 is going to give me -3; so I know that the right combination would be to have a 2 here and my 5 right here, with a negative sign.0993

All right, now I see that I have everything factored as far as it can be factored.1012

So, up here, it states that I need to factor these before I can simplify the product or quotient.1021

So, my next step, after I have factored, is to go ahead and do the simplification by canceling out factors that are common to the numerator and denominator.1029

OK, I see that I have an (x + 2) here; I also have an (x + 2) in the denominator.1043

I have an (x - 7) in the numerator and an (x - 7) in the denominator.1050

I have no other common factors, so I am going to multiply what is left in the numerator and in the denominator.1055

I can simplify this a bit more, because 2 times 5 is 10; that is going to give me 10 times (x - 5), divided by (x + 7) (x + 2).1070

And then, I double-check again at the end, to make sure I haven't missed any factors and there are no common factors.1080

It is good practice to simplify, of course, before you multiply, to make your multiplication easier.1086

But then, at the end, go back and double-check and make sure that there are no common factors--that you can't do any more canceling out.1092

A complex fraction is a rational expression whose numerator or denominator contains a rational expression.1103

Let's step back again and look at complex fractions when we are just talking about numbers, not yet working with polynomials.1110

An example of a complex fraction, just with numbers, would be something like 3/7, all that divided by 5/2.1117

Here I have a fraction, and the numerator of that fraction, and the denominator, are fractions; it is a complex expression.1126

I could also have something like 1/2 divided by 4; and here, just the numerator is a fraction.1134

That is still a complex expression: if either the numerator, the denominator, or both involve fractions, you have a complex expression.1143

So, looking at how to work with complex fractions that are algebraic complex fractions1152

(meaning that they involve rational expressions), let's look at this example.1160

(2x + 6)/(x2 - 8x + 16), divided by (x2 - 9)/(x2 - 2x - 8):1166

now, this fraction bar, as you know, means "divided."1186

So, I have a complex fraction where both the numerator and the denominator are comprised of rational expressions.1190

To simplify a complex fraction, write it as a division expression and simplify.1199

So, I am looking at the main fraction bar and realizing that all this is saying is to divide.1204

So, I am going to rewrite this as (2x + 6), divided by (x2 - 8x + 16),1210

all of this--this numerator--being divided by the denominator.1223

At this point, I am not doing anything except writing it out in a form that is more recognizable and easier to work with.1229

Once I have done this, this just becomes division with rational expressions.1239

And I know that, in order to divide one rational expression by another, I am going to multiply the first one by the inverse of the second...1243

times x2 - 8x + 16...I am taking...this right here is x2...1256

the inverse of the second, minus 2x, minus 8, divided by x2 - 9.1275

OK, so again, what I have done is taken the first rational expression, and I multiplied it by the inverse of the second, (x2 - 2x - 8)/(x2 - 9).1288

This is now just a multiplication problem; and recall that the next step is going to be to go ahead and factor, simplify, and multiply.1305

I can factor this out to 2 times (x + 3), divided by...well, this is going to be x, and this is x.1318

I have a positive sign here and a negative here, and what that is telling me is that I have a negative and a negative,1329

because when I multiply those two out, I am going to get a positive here; but when I add them, I will end up with a negative.1336

And this is actually (x - 4) (x - 4); I recognize that as (x - 4)2.1343

Here, I have x times x, and I have a negative here, which tells me I am going to have a positive sign here and a negative here.1354

The factors of 8 are 1 and 8, and 2 and 4.1366

And I am looking for factors of 8 that add up to -2.1371

So, I am going to look at these two, because they are close together.1375

And I know that, if I take -4 and positive 2, and I add those up, I am going to get -2.1378

So, I am going to get x2 - 4x + 2x, and -4x + 2x is going to give me -2x, and then -8 over here.1394

Down here, you probably recognize that this is (x - 3) (x + 3).1406

All right, so now I am going to look for common factors.1415

And the common factors that I have are going to be (x + 3) (I have an x + 3 here--I am going to cancel those out);1419

over here, I have (x - 4), and here I have an (x - 4), so those are going to cancel out.1438

So, I am looking around--are there any other common factors?--and there are not.1456

So, I am going to just multiply the numerator and the denominator: that is 2 times (x + 2), divided by (x - 4) times (x - 3).1459

Double-check at the end: do I have any common factors?1477

I do not, so this is in simplest form.1479

So again, I started out with a complex fraction; my first step was to simply rewrite that as a division problem.1482

So, I had the numerator right here, divided by the denominator, which is right here.1489

Then, I changed that to a multiplication problem; and with a multiplication problem,1498

I am going to set it up as the first rational expression (unchanged), times the inverse of the second rational expression.1506

Therefore, x2 - 2x - 8 becomes the numerator, and x2 - 9 becomes the denominator.1516

Once I have that, the next thing is to treat it like a regular multiplication problem, which is what it became.1526

So, I factored this out to (x + 3) times 2, factored the denominator, and did the same thing for the second one;1531

and then I looked through and saw that I had several common factors, so I went ahead and canceled those out.1541

OK, so that was handling complex fractions; and it just involved putting together what we have learned about division and multiplication of rational expressions.1548

Let's look at our first example: the first example just asks me to simplify a rational expression.1557

And the first step is going to be factoring.1563

I have 16 - x2: this is going to give me (4 + x) (4 - x).1567

And we are used to working with the opposite situation, x2 - 16, but it is a very similar idea.1576

You can check that this is 4 times 4 (is 16), and then I multiply; that is -4x + 4x (those drop out), and then I get x times -x to give me -x2.1583

So, that is factored correctly.1596

In the denominator, I have an x here, times an x here.1598

I have a negative sign in front of the constant, so I am going to have a plus here and a minus here.1603

Now, I need to just think about factors of 12.1609

And I need to look for factors of 12 that add up to -1.1614

And -1 is small, so I am going to look for factors that are close together; and that would be 3 and 4.1620

I am going to make the larger one negative, and the 3 positive, because I want to end up with a negative; and these equal -1.1626

I know that my correct factorization of the denominator would be (x + 3) (x - 4).1634

Now, to compare what is going on more easily, I am going to rewrite the numerator, putting the x's first.1643

I have (x + 4); here I am going to have (-x + 4).1650

This allows me to compare the factors, now that everything is in the same order--the terms within the factors are in the same order.1656

I can see here that I obviously don't have a common factor with this, and I don't have one with this.1666

But if I look at these, they are pretty close.1670

I again have the situation we discussed, where the signs are opposite.1673

I have a negative here; I have a positive here; I have a positive here; I have a negative here.1680

And remember: the way we handle that is to factor a -1 out from either this one or this one--you can choose either one.1689

I am going to go ahead and factor a -1 from the numerator--factor the -1 out from right here.1697

So, I am going to pull this out, and this is going to give me x - 4.1710

I now see that I have a common factor; so I can simplify by canceling out that common factor, leaving behind (x + 4) times -1...1716

I will pull that out in front...divided by (x + 3).1735

And just to write it even in a more standard form, instead of putting -1, I will pull the negative sign out in front.1740

And this becomes -(x + 4) divided by (x + 3).1746

OK, the first step is always factoring; then canceling out common factors--if you don't have a common factor,1753

see if factoring out a -1 from the numerator or denominator would give you a common factor (which it did).1760

And then, you can cancel those out.1767

And then, double-check at the end: I have no common factors, so I am done.1769

Example 2: we are asked to simplify here, and it is multiplication.1775

The first step is to factor; so in the numerator of the first rational expression,1780

I have a negative sign here; so I am going to put a + here and a - here.1788

Factors of 15 are 1 and 15, 3 and 5; I want those to add up to 2 (for 2x), so I know that these are too far apart; I am going to focus on this.1795

I want a positive sign here, so I am going to make the larger number positive to get 5 - 3 = 2.1810

So, I know I want the 5 here and the 3 here.1818

In the denominator, I have, again, the negative sign here; so it is (x + something) (x - something).1826

Factors of 10 are 1 and 10, 2 and 5.1835

I want them to add up to 3x, and so, again, I am going to look for the factors that are close together, which are going to be 2 and 5.1839

And I want a positive sign, so I am going to make 5 positive.1847

And those do add up to 3, so this is going to give me (x + 5) (x - 2) times...this time, I have a common factor of 4.1852

In the denominator, I have a common factor of 3.1863

Now, I have factored everything out; it is time to just cancel out common factors before I multiply.1867

(x + 5) and (x + 5) are common factors; (x - 3) and (x - 3) are also common factors.1875

(x - 2) and (x - 2)...so there are a lot of common factors.1884

And all I have left is 4/3; so what looked like a very complicated expression, actually, is just equivalent to 4/3.1888

Here we are asked to simplify, and this involves division.1900

So, when I see division, the first thing I do is rewrite it as multiplication: the first rational expression times the inverse of the second.1903

This becomes the numerator: x2 + 12x + 36, divided by 3x2 + 7x - 6.1915

OK, start factoring: and this factoring is a little more difficult, because the leading coefficient is not 1--it is 2.1926

I know I am going to have 2x here and an x here; I also know that I have a negative here.1936

So, one is going to be positive; one is going to be negative; but I don't actually know if the positive goes with the 2x or with the x.1942

I am going to have to do some trial and error up here.1948

I am trying to factor out 2x2 + 5x - 3; and I know that I want the middle term to add up to 5x.1950

Factors of 3--there are not many, so this makes it a lot simpler--it is just 1 and 3.1959

So, trial and error: let's first put the 2x with the positive sign, and let's try the x with the negative sign.1965

I am going to go ahead and try 1 here and 3 here; and this is going to give me -6x + x, so that is going to add up to -5x.1976

I know that that is not correct, because the sign is wrong.1986

But I have the right number, 5; it is just that the sign is wrong.1989

So, I am going to try this again, but putting the negative sign with the 2x factor.1992

I am going to try this again: the outer term gives me 6x; the inner term is -x; and that equals 5x.2000

So, I know that this is the correct factorization.2007

The bottom is much simpler to factor: you will recognize that as (x - 6) (x + 6).2016

And when you multiply this out, the middle term would drop out, and you will just end up with x2 - 36.2023

Here, I have x, and an x, and I have a positive sign here and here; so I am going to end up with a + and a +.2030

Factors of 36 (and I want these to add up to 12): I have 1 and 36, 9 and 4, and 6 and 6.2042

I know that these two add up to 37, so that is not correct; this adds up to 13; 6 + 6 is 12, so these are the correct terms within the factor.2058

OK, now this one, again, is a little bit more complicated, because the leading coefficient is not 1.2074

So, I have 3x and x, and I am dealing with a negative sign, so I have some kind of combination of positive and negative.2081

And I have 3x here, and I have x here; and I am thinking about the factors of 6 that are going to add up to positive 7.2096

So, I have 1 and 6, and 2 and 3; since I have a 3 that is going to amplify what I multiply, I am going to stay away from this 6 for right now.2122

I am going to work with 2 and 3; so let's try (3x + 2) and (x - 3).2135

This is going to give me...the first terms are 3x2; outer terms are going to be -9x.2142

The inner terms are + 2x, so that is going to add up to -7x.2150

What this is telling me is that I have the right combination, but the wrong signs.2156

So, I am going to try this again: (3x - 2) (x + 3).2160

Now, I multiply this out: 3x2...the outer terms give me 9x; the inner terms, -2x; and this gives me 7x.2166

So, this is my correct factorization; so you see, at this point in the course, we are using factoring as a tool to solve problems.2177

So, you really need to have the factoring down; and you can go ahead and review that in the Algebra I videos, if you need to,2185

especially for the more complicated problems, when the leading coefficient is something other than 1.2191

OK, so I have the factorization here: (3x - 2) and (x + 3).2198

So, I have this all factored out; now is the easy part--I just get to cancel out like factors.2205

I have (2x - 1); there is no common factor; I have (x + 3)--that is a common factor; I am going to cross that out and cancel it out.2211

I have (x + 6); there is a common factor--get rid of that; and one more (x + 6)--no common factor.2222

(x - 6) is no common factor; (3x - 2)--that is it; I am done with my common factors.2232

I multiply what is left over, which is (2x - 1) and (x + 6), divided by (x - 6) (3x - 2).2238

OK, so this took longer; it was more complicated; but it is really the same technique.2249

Start out by rewriting division as multiplication of the first rational expression, times the reciprocal of the second.2254

Factor; cancel common factors; and then, at the end, check and make sure you haven't missed any common factors.2261

Here we have a complex fraction involving rational expressions in both the numerator and the denominator.2271

The first step is always to rewrite this as a division problem.2277

Remember that the fraction bar is telling me to divide; so it is the entire numerator, divided by the denominator.2286

Recall that, in order to divide, we take the first rational expression (the first fraction) and multiply it by the inverse of the second.2300

At this point, we are just working with a multiplication problem.2313

The next step is to factor, and then simplify.2317

So, for the numerator here, there is no way to factor that out; let's go to the denominator.2321

I have a negative here, so this becomes a plus, and this is a minus.2331

And factors of 6 are 1 and 6, 2 and 3; I want factors that add up to -1.2336

These two are close together; so if I take -3 + 2, that equals -1.2345

So, my 3 is going to go by the negative; my 2 is going to go by the positive.2352

Now, to factor x2 + x - 12...again, I have a negative sign, so I am going to do plus and minus.2359

Factors of 12 are 1 and 12, 2 and 6, 3 and 4.2369

And since I want these to add up to 1, I am going to pick the two that are close together, which are 3 and 4.2375

And I want this to be positive, so I am going to make the 4 positive and the 3 negative.2381

4 - 3 is 1, so it is (x + 4) (x - 3).2386

In the denominator, I have a greatest common factor of 4, so this becomes 4(2x - 3).2396

So, I did my factoring; and I see here that I have (2x - 3) in both the numerator and the denominator; those cancel.2405

I don't have any common factor with (x + 4); I do have a common factor with (x - 3); and this has no common factor.2413

Now, I multiply what is left behind: (x + 4) divided by...I have a 4 here and an (x + 2) there.2424

Finally, I double-check to make sure that there are no common factors that were missed.2433

I cannot simplify any more, so this complex fraction is now in simplified form.2437

That concludes this lesson on rational expressions, on multiplying and dividing.2447

And I will see you next lesson on Educator.com!2451

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