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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Adding and Subtracting 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 (2)

0 answers

Post by Dr Carleen Eaton on May 24, 2012

Julius, if I understand the question, at 15:00, it is easier to factor out the 2. Otherwise your LCD is (x-3)(x 1)(2x-6). No reason it shouldn't work but it is a little more complicated and to get the most simplified form you'll have to do a difficult factoring of a cubic equation at the end.

0 answers

Post by julius mogyorossy on May 21, 2012

Dr. Eaton, in the example you gave us I did not factor out the two first, I just factored it, did I do it the wrong way, it seems so, when I substitute 1 in for x it seems I get a different solution than you, but it really seems the way I did it should work 2, if you must factor out the 2 first, why so please?

Adding and Subtracting Rational Expressions

  • To add or subtract: first, find the LCM of the denominators. Then adjust the numerator and denominator of each fraction so that its denominator is the LCM. Finally, add or subtract the numerators and simplify the result.
  • Most of the time, the result will not simplify.

Adding and Subtracting Rational Expressions

Subtract: [1/2x] − [2/10x]
  • List the prime factors of 2x and 10x to find the LCD.
  • 2x =
  • 10x =
  • 2x = 2*x
  • 10x = 2*5*x
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = 2*5*x = 10x
  • Change each rational expression into an equivalent expression with the LCD.
  • [1/2x] − [2/10x] = ( [5/5] )[1/2x] − [2/10x] =
  • [5/10x] − [2/10x] =
  • [(5 − 2)/10x] =
[3/10x]
Add: [1/7x] + [2/x]
  • List the prime factors of 7x and x to find the LCD.
  • 7x =
  • x =
  • 7x = 7*x
  • x = x
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = 7*x = 7x
  • Change each rational expression into an equivalent expression with the LCD.
  • [1/7x] + [2/x] = [1/7x] + [2/x]( [7/7] ) =
  • [1/7x] + [14/7x] =
  • [(1 + 14)/7x] =
[15/7x]
Add: [10/(xy2)] + [5/(y2)]
  • List the prime factors of xy2 and y2 to find the LCD.
  • xy2 =
  • y2 =
  • xy2 = x*y*y
  • y2 = y*y
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = x*y*y = xy2
  • Change each rational expression into an equivalent expression with the LCD.
  • [10/(xy2)] + [5/(y2)] = [10/(xy2)] + [5/(y2)]( [x/x] ) =
  • [10/(xy2)] + [5x/(xy2)] =
[(10 + 5x)/(xy2)] =
Subtract: [9/(a3)] − [7/a]
  • List the prime factors of a3 and a to find the LCD.
  • a3 =
  • a =
  • a3 = a*a*a
  • a = a
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = a*a*a = a3
  • Change each rational expression into an equivalent expression with the LCD.
  • [9/(a3)] − [7/a] = [9/(a3)] − [7/a]( [(a2)/(a2)] ) =
  • [9/(a3)] − [(7a2)/(a3)] =
[(9 − 7a2)/(a3)] =
Add: [2/(3x + 6)] + [5/(x + 2)]
  • List the prime factors of 3x + 6 and x + 2 to find the LCD.
  • 3x + 6 =
  • x + 2 =
  • 3x + 6 = 3*(x + 2)
  • x + 2 = (x + 2)
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = 3*(x + 2)
  • Change each rational expression into an equivalent expression with the LCD.
  • [2/(3x + 6)] + [5/(x + 2)] = [2/(3(x + 2))] + [5/(x + 2)]( [3/3] ) =
  • [2/(3(x + 2))] + [15/(3(x + 2))] =
  • [(2 + 15)/(3(x + 2))] =
  • [17/(3(x + 2))]
  • or
[17/(3x + 6)]
Subtract: [7/(2x − 8)] − [2/(x − 4)]
  • List the prime factors of 2x − 8 and x − 4 to find the LCD.
  • 2x − 8 =
  • x − 4 =
  • 2x − 8 = 2*(x − 4)
  • x − 4 = (x − 4)
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = 2*(x − 4)
  • Change each rational expression into an equivalent expression with the LCD.
  • [7/(2x − 8)] − [2/(x − 4)] = [7/(2(x − 4))] − [2/((x − 4))]( [2/2] ) =
  • [7/(2(x − 4))] − [4/(2(x − 4))] =
  • [(7 − 4)/(2(x − 4))] =
  • [3/(2(x − 4))]
  • or
[3/(2x − 8)]
Add: [2x/(x + 1)] + [x/(4x + 4)]
  • List the prime factors of x + 1 and 4x + 4 to find the LCD.
  • x + 1 =
  • 4x + 4 =
  • x + 1 = (x + 1)
  • 4x + 4 = 4(x + 1)
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = 4*(x + 1)
  • Change each rational expression into an equivalent expression with the LCD.
  • [2x/(x + 1)] + [x/(4x + 4)] = ( [4/4] )[2x/((x + 1))] + [x/(4(x + 1))] =
  • [8x/(4(x + 1))] + [x/(4(x + 1))] =
  • [(8x + x)/(4(x + 1))] =
  • [9x/(4(x + 1))]
  • or
[9x/(4x + 4)]
Add: [7x/(x2 − 16)] + [2/(x + 4)]
  • List the prime factors of x2 − 16 and x + 4 to find the LCD. Factoring may be required.
  • x2 − 16 =
  • x + 4 =
  • x2 − 16 = (x + 4)(x − 4)
  • x + 4 = (x + 4)
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = (x + 4)(x − 4)
  • Change each rational expression into an equivalent expression with the LCD.
  • [7x/(x2 − 16)] + [2/(x + 4)] = [7x/((x + 4)(x − 4))] + [2/(( x + 4 ))]( [((x − 4))/((x − 4))] ) =
  • [7x/((x + 4)(x − 4))] + [(2(x − 4))/(( x + 4 )(x − 4))] =
  • [(7x + 2(x − 4))/((x + 4)(x − 4))] =
  • [(7x + 2x − 8)/((x + 4)(x − 4))]
  • [(9x − 8)/((x + 4)(x − 4))]
  • or
[(9x − 8)/(x2 − 16)]
Add: [4x/(x − 1)] + [2/(x + 4)]
  • List the prime factors of x2 − 16 and x + 4 to find the LCD. Factoring may be required.
  • x2 − 16 =
  • x + 4 =
  • x2 − 16 = (x + 4)(x − 4)
  • x + 4 = (x + 4)
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = (x + 4)(x − 4)
  • Change each rational expression into an equivalent expression with the LCD.
  • [7x/(x2 − 16)] + [2/(x + 4)] = [7x/((x + 4)(x − 4))] + [2/(( x + 4 ))]( [((x − 4))/((x − 4))] ) =
  • [7x/((x + 4)(x − 4))] + [(2(x − 4))/(( x + 4 )(x − 4))] =
  • [(7x + 2(x − 4))/((x + 4)(x − 4))] =
  • [(7x + 2x − 8)/((x + 4)(x − 4))]
  • [(9x − 8)/((x + 4)(x − 4))]
  • or
[(9x − 8)/(x2 − 16)]
Add: [3x/(x2 + x − 20)] + [2/(x + 5)]
  • List the prime factors of x2 + x − 20 and x + 5 to find the LCD. Factoring may be required.
  • x2 + x − 20 =
  • x + 5 =
  • x2 + x − 20 = (x + 5)(x − 4)
  • x + 5 = (x + 5)
  • Use each prime factor the greatest number of times it appears in each of the factorization
  • LCD =
  • LCD = (x + 5)(x − 4)
  • Change each rational expression into an equivalent expression with the LCD.
  • [3x/(x2 + x − 20)] + [2/(x + 5)] = [3x/((x + 5)(x − 4))] + [2/(x + 5)]( [((x − 4))/((x − 4))] ) =
  • [3x/((x + 5)(x − 4))] + [(2(x − 4))/((x + 5)(x − 4))] =
  • [(3x + 2(x − 4))/((x + 5)(x − 4))] =
  • [(3x + 2x − 8)/((x + 5)(x − 4))]
  • [(5x − 8)/((x + 5)(x − 4))]
  • or
[(5x − 8)/(x2 + x − 20)]

*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

Adding and Subtracting 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
  • Least Common Multiple (LCM) 0:27
    • Examples: LCM of Numbers
    • Example: LCM of Polynomials
  • Adding and Subtracting 7:55
    • Least Common Denominator (LCD)
    • Example: Numbers
    • Example: Rational Expressions
    • Equivalent Fractions
  • Simplifying Complex Fractions 21:19
    • Example: Previous Lessons
    • Example: More Complex
  • 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

Transcription: Adding and Subtracting Rational Expressions

Welcome to Educator.com.0000

Today, we are going to be working on adding and subtracting rational expressions.0002

And recall that rational expressions are algebraic fractions; so this is more complicated than multiplying or dividing rational expressions.0006

But we are just going to apply the same techniques that we used to add or subtract fractions involving numbers to add or subtract rational expressions.0017

Let's start out by reviewing some concepts from working with fractions.0027

Recall the idea of the least common multiple, because we are going to use that to get a common denominator.0032

As you know, if you have a common denominator with your fractions (such as 2/4 + 1/4), it is very easy to add.0038

You just add the numerators (2 + 1 is 3) and use the common denominator.0046

It is the same with rational expressions: if you already have a common denominator, add the numerators0052

and use the common denominator as the denominator of the rational expression that you end up with from adding or subtracting.0058

However, just like with fractions, when you don't have a common denominator,0069

you have to find a common denominator and convert to equivalent fractions before you can add or subtract.0073

Let's talk about an example using numbers: if I want to add 1/10 and 2/45, you might just look at this,0081

know what the common denominator is, quickly convert it, possibly even in your head, and add it.0091

But I am going to bring out each step of that consciously, so we can use that to work with adding and subtracting rational expressions.0096

Let's look at the denominators: let's find the least common multiple of these two numbers, 10 and 45.0105

In order to find a common multiple, you need to factor.0113

And whether we are working with a number, or if the denominator is a polynomial (as in a rational expression),0118

the idea is still that you have to factor.0123

So, let's factor this out: and I get 2 times 5 here; for 45, I might say it's 9 times 5, but I can go farther with my factoring.0127

9 factors into 3 times 3; I could also rewrite this as 32 times 5.0137

So, I have 2 times 5 for 10, and then I have 32 times 5 as my prime factorization for 45.0144

The least common multiple is going to be the product of the unique factors that I see here,0152

to the highest power that they appear for any one number (or, when we are talking about polynomials, for any one polynomial).0161

So, the unique factors: 2 is a factor, and I am going to take the product of that and the other factors I see.0170

5 is here once only; and it is here once only; so the greatest number of times that 5 appears in any one set of factors is once.0179

So, I am not going to write it twice: I am just going to write it once.0188

I am done with this one; here I still have another factor that isn't accounted for; I have a 3.0192

But it appears twice, so I am going to put 32.0197

Now, if I had 5 twice up here, then I would have written 52; I only had it once.0201

All right, this equals...2 times 5 is 10, times 32...so 10 times 9 is 90.0206

The least common multiple of these two is 90; and I am going to end up using that as my common denominator.0215

But as you know, I can't just change this to 1/90 and add that to 2/90, because those fractions, then, won't be the same.0221

So, we are going to talk about, once you get the common denominator that you are going to use, what to do with the rest of the fraction.0228

And we will talk about that in a minute; but first, let's apply this concept to finding the least common multiple of polynomials.0234

If I was working with something like 1 over x2 plus 7x, plus 14, and I wanted to add that0241

to 1 over 2x2 - 2x - 12, then I need to find the common multiple of these two denominators.0250

So, to find the LCM, I am going to factor out both of these denominators, just the way I did up here with the numbers.0260

OK, so this is going to give me x, and then x; I have all plus signs, so I know that these are going to be positive.0272

Factors of 14...actually, let's make this a 10...let's go ahead and make that a 10 for much easier factorization.0282

This is going to be factors of 10: 1 and 10, and 2 and 5; and I need them to add up to 7, and 2 + 5 = 7.0298

OK, so this is factored out as far as I can go.0313

Now, down here, I am aware that I have a common factor of 2;0319

so I am going to pull that out to get 2(x2 - x...and that is going to give me - 6).0325

Now, work on factoring this: 6 has factors that are 1 and 6, and 2 and 3.0341

I have an x here; I have an x here; and since this is a negative, I have to have a positive and a negative.0351

And I want them to add up to -1, so I need to find factors that are close together.0359

And I need to make the larger factor negative, so that this will add up to -1.0364

Let's try 2 - 3; that equals -1; so I am going to make the 2 positive and the 3 negative.0368

OK, now I factored out both denominators; and this is going to then give me my least common multiple.0377

I am going to look for each factor that appears.0386

And I see an x + 2; it appears here once, and it only appears here once.0389

So, the highest power to which it appears is just 1, so it is x + 2.0394

I also have an x + 5; it appears once, and I am going to put it once.0401

This is done; I accounted for this; I also need an x - 3.0412

Therefore, the least common multiple of these denominators is going to be (x + 2)(x + 5)(x - 3)...0416

and let's not forget about the 2; the 2 needs to come along as well, so 2(x + 2)(x + 5)(x - 3).0427

If I had a situation where I factored something out, and I got (x + 3)(x + 3)(x - 1), and then I had down here0438

(x + 3)(x - 5), then the LCM would be...I have an (x + 3), but it appears twice in this polynomial, so it would be (x + 3)2.0448

And then, I would have my x - 1 and x - 5.0461

So, the least common multiple will allow you to find a common denominator when you are adding or subtracting rational expressions.0468

To add or subtract, first find a common denominator.0476

The LCM of the denominators is the smallest value that we can use as a common denominator.0479

And we are going to refer to that as the least common denominator.0484

The least common denominator is just the LCM of the two denominators.0487

OK, so let's go back to our example with numbers.0492

And I had 1/10, and I asked you to add 2/45.0497

We factored out 10 to get 2 times 5; we factored out 45 to get 32 times 5.0505

The LCM...a 2 appears up here; 5 appears, at most, once; and 3 appears twice, so it is 32.0516

This equals 90; now, I am going to use this 90 as the least common denominator; so, the least common denominator for these two fractions is 90.0528

But I can't simply say, "OK, I am going to make this 1/90 and 2/90"; those won't be the same fractions.0538

I need to convert these to equivalent fractions, but with my new denominator.0544

Again, this is something you just know how to do; but we need to bring it out and look at each step.0548

So, let's talk about converting 1/10: if I rewrite this as 1/(2 times 5), it is factorization.0552

That allows me to look at the LCM and determine what I am missing--what factor am I missing?0559

I have a 2; I have a 5; but I am missing the 32.0566

The thing is that I can't just multiply the denominator alone; what I am allowed to do0572

is multiply both the numerator and the denominator by the same number, because this is just 1--0577

this cancels out to 1--and I am allowed to multiply by 1.0584

So, this is 1 times 9; this is 2 times 5 is 10, times 9 is 90; and of course, that simplifies to 1/10, so I know I have a fraction that still has the same value.0587

2/45 can be written as 2 over (32 times 5); what is missing from this denominator?0602

Well, I have a 5; and I have a 32; I look down here--the 2 is missing.0611

So, I need to multiply both the numerator and the denominator by 2.0621

2 times 2 is 4, over 9 times 5 is 45, times 2...so that is 4/90.0627

Now, I can add these two: I have 9/90 + 4/90; that is 13/90.0633

We are doing the same thing when we add rational expressions or subtract rational expressions.0641

We are going to take the same steps: we are going to find the least common multiple, and that will be our common denominator.0646

Then we are going to convert to fractions that are equivalent to the original fractions, but with the new denominator.0651

And then, we can add or subtract those once they have the same denominator.0658

OK, looking at an example using rational expressions: here is my first rational expression, and I am going to add that to (x + 4)/(2x - 6).0663

My first step is to find the LCD: factor out the denominators, and then find the least common multiple to use that as the denominator.0678

So, I have my two denominators: this factors into, since it has a negative, (x + something) (x - something).0692

The only factors of 3 that I am going to have to work with are 1 and 3.0703

Since I want to end up with a negative 2x, I am going to make the larger number negative: 1 - 3 is -2.0708

So, I am going to make this a 2: (x + 2) (x - 3), and (x + 1) (x - 3); so that is0717

x2 - 3x + 1x (so that is -2x), and then the last terms multiply out to -3.0728

OK, and this should be 2x - 6, so I am going to look at the common facto that I have, which is 2.0738

So, this factors out to 2(x - 3); the LCM is the product of the factors of both of these.0756

I have (x + 1), and that only appears once; that is the highest power to which it appears in any of these polynomials.0771

So, I am just going to put (x + 1).0780

I also have an (x - 3), and that is present once here and once here, so I represent it once.0782

Down here, I also have a 2; I am going to pull that out in front; the LCM is 2(x + 1) (x - 3).0789

And I am going to use it as the least common denominator.0797

We found the common denominator; next we need to find the equivalent fractions, using that common denominator (using the LCD as their denominator).0805

Once I have done that, all I have to do is add or subtract the numerators of the fractions, and then simplify.0814

Let's continue on with the example I started in the previous slide, which was 1/(x2 - 2x - 3) + (x + 4)/(2x - 6).0822

This factored out to 1/(x - 3) (x + 1).0839

This, you recall, factored into 2(x - 3).0859

So, I factored these out; I haven't done anything else with them yet--I have left them the same, but I have factored them out.0869

So, I have them factored: the LCD is going to be the product of the unique factors.0874

Remember, I had an (x - 3) as part of that; I had an (x + 1), which appears here; I already have my (x - 3) accounted for; and I need a 2.0882

So, I am going to rewrite this with a 2 out in front; and I am going to use my least common multiple as my denominator, my LCD.0893

2 times (x - 3) times (x + 1)...0904

The hard part can be converting to equivalent fractions.0909

You just have to be careful and make sure that you multiply the numerator and the denominator by the same thing.0912

Otherwise, you will not end up with an equivalent fraction.0917

To convert to equivalent fractions is my next step.0922

I am going to take this first fraction, and I am going to say, "OK, what is lacking?"0935

I want to turn this denominator into this.0939

I have my (x - 3); I have my (x + 1); but I am missing a 2.0944

Therefore, I am going to multiply both the numerator and the denominator by 2/2, which is just multiplying this fraction by 1.0949

The second fraction: to turn this denominator into what I have here, I have to figure out what I am missing.0960

I have a 2; I have an (x - 3); I am missing the (x + 1).0974

So, I am going to multiply both the numerator and the denominator by (x + 1).0978

Again, this is just multiplying by 1.0984

This is going to give me...1 times 2 is 2, over (x - 3) (x + 1) (2)...and I am pulling the 2 out in front.0988

Here, I am going to add that; and in the numerator, I am going to have (x + 4) (x + 1) for the second fraction, for the numerator.1006

In the denominator, I am going to have that same common denominator: 2/(x - 3)(x + 1).1022

So, I have just multiplied the numerator and the denominator by whatever was missing from the denominator to form the LCD.1031

Now that I have a common denominator, I just add or subtract the numerator of these fractions.1041

So, since I am adding, this is going to become 2 + (x + 4)(x + 1), all over my common denominator.1047

In the same way, if I am adding 3/4 and 5/4, I just add the numerators and put them over the common denominator.1065

The same idea here--I added the numerators and put them over the common denominator.1076

Finally, I am going to simplify: I am going to need to multiply this out, and this is going to give me 2 +...let's work up here:1084

(x + 4) times (x + 1): using FOIL, this is going to give me x2, and then the outer term is going to be + x;1094

then inner term is going to be 4x, so that is 5x; and then, the last term is going to be 4.1106

So, this is going to give me x2 + 5x + 4, all over 2 times (x - 3)(x + 1).1111

So, I am working right over here, just to do that multiplication.1129

(x - 3) times (x + 1)...and then we are going to have to multiply all of that by 2.1133

That is going to give me...actually, we are going to go ahead and leave that in factored form, because that will make it easier to simplify.1141

Let's just go ahead and leave that one in factored form.1151

So, the top gives me x2, and...I have 5x; I leave that alone; and this is 2 + 4, to give me 6, all over this.1157

Now, at the top, I went ahead and multiplied this out, so I could add it to this.1174

For the denominator, I didn't need to do that, because it is already factored out; and you will see why in a second.1179

OK, can I factor this? Well, let's go ahead and see.1185

I have x and x, and right here, I have positives; so I am going to make that a + and make that a +.1189

6...what are my factors? 1 and 6, 2 and 3.1201

So, I am looking for factors of 6 that are going to end up giving me 5; and I can see that 2 + 3 = 5, so this is going to be (x + 3) (x + 2).1207

That is x2 + 2x + 3x (is 5x) + 6.1219

OK, now you can see why I just left the denominator in factored form.1228

I look here, and I see if I can simplify--are there any common factors?1232

And there are actually not--I don't have any common factors in the numerator or denominator.1235

And it is perfectly fine to leave the expression like this.1239

You might want to go on and do something else with it, and then it is already factored.1243

For something like we had in the numerator, where we ended up with 2 plus all of this, we need to actually combine that.1246

So, I needed to multiply this out, add, and then factor.1254

This was already factored out; so I left it, and this was my final answer.1258

So again, I found the LCD of these two by factoring.1262

This was the LCD; I converted the first and the second fractions into equivalent fractions, and then I just added and tried to simplify.1268

OK, we have talked before about simplifying complex fractions, but this time1279

we are going to talk about complex fractions that also may require you to add or subtract in the numerator or denominator.1284

So, first I want to explain the difference between what we did in a previous lesson and what we are going to do now.1290

In the previous lesson, we talked about things like this, rational expressions that are complex fractions: (3xy/(x2 - 16))/ ((2x + 1)/xy).1297

So, the steps for this were to rewrite this as division, because I know that this fraction bar is telling me to take the numerator and divide it by the denominator.1314

Once you got to this point, you recognized that this is simply the first rational expression, times the reciprocal of the second.1330

And I am not going to do the whole multiplication right now; I just wanted to show you the setup on that.1352

So, if I had a complex fraction, such as this, I just took the numerator, set it as divided by the denominator,1357

and then converted it to multiplication of the first rational expression times the inverse of the second.1364

OK, what we are talking about here is actually different.1373

It is more complex, and requires one more step before you can go from here to here.1377

Let's say I have something like this: (1/x + 2/y)/(3/x - 1/y).1385

Now, not only do I have a complex fraction (I have a situation where I have a fraction in the numerator and a fraction in the denominator),1400

but I have a sum and/or a difference (here I have a sum; here I have a difference).1408

I have a fraction up here that I am adding to another fraction; I have a fraction down here that I am subtracting from another fraction.1413

Up here, I just had one fraction; I could just go ahead and simplify by multiplication; down here, I just had one fraction.1419

So, the difference here is: we are talking about simplifying complex fractions1426

in which you need to add or subtract in the numerator or the denominator of the complex fraction.1430

What you are going to do is handle these separately.1437

You are going to handle the numerator; you are going to handle the denominator; and then you are going to put it all together.1439

So, to simplify a complex fraction, add or subtract the fractions in the numerator and the denominator separately, and then simplify.1443

I am going to start out with the numerator.1451

In the numerator, I have 1/x + 2/y; I need to find a common denominator.1457

And since all I have in the denominator is an x, and all I have in the denominator over here is a y, then my LCD (or my LCM) is going to be xy.1464

Now, I need to convert these to equivalent fractions.1477

I see that what I am lacking from this denominator is a y, so I need to multiply this times y/y.1485

Now, I have an xy in the denominator, and I multiplied the numerator by the same thing, so that I end up with equivalent fractions.1495

Over here, I want to get the LCD of xy; what I am lacking in the denominator is an x.1501

So, I am going to multiply both the numerator and the denominator by x.1508

This is going to give me y/xy, plus 2x/xy.1513

Now, I can add those, because they have the same denominator.1526

And it is just going to be (y + 2x)/xy.1528

Now, I go back here, and what I have in the numerator now is (y + 2x)/xy.1534

I no longer have the sum, where I have two separate fractions: I have one fraction in the numerator.1546

OK, denominator: the same thing--the denominator is 3/x - 1/y.1556

Again, I have an LCD down here--my LCD is just going to be xy, because this is the only factor here; this is the only factor here.1577

I need to convert these to equivalent fractions, so what I am going to do is say, "All right, what am I lacking from this?"1588

I am lacking a y in the denominator, so I am going to multiply this times y/y.1598

And I am subtracting that from 1/y, and what I am lacking in the common denominator is an x.1606

So, I am going to multiply this times x/x; this is going to give me 3y/xy - x/xy.1612

I now have a common denominator: this gives me (3y - x)/xy: this is my denominator, (3y - x)/xy.1629

That was the hard part: once you get to here, you are working with this situation.1643

I could handle this by rewriting this as a division problem: (y + 2x)/xy--I am going to take that;1649

it is going to be like this first rational expression; and I am going to say "divided by" this whole thing.1657

Once you get to that point, you use our usual method of taking the first rational expression and multiplying it by the inverse of the second.1663

This is the difficult step: and the way to really look at this is to handle the numerator and the denominator separately.1672

Your goal is to get the numerator to look like this (a single fraction) and the denominator to look like this.1678

This is an addition problem with a rational expression; this is an addition problem with a rational expression.1684

And once you take care of the numerator and you take care of the denominator, then you can proceed as usual.1693

This is a complex problem: it just has a lot of steps, and you just need to take it one at a time and keep track of what you are working with.1702

OK, with the examples, we are going to start out just finding the LCM; but this time, it is going to be of three polynomials.1712

So, when we find the LCM, no matter if it is 2, 3, 4, or more, we need to factor.1718

I am going to factor the first polynomial, the second polynomial, and the third polynomial.1728

OK, so this is x; and I have a negative sign here, but I have a positive sign here.1743

That clues me into the fact that I have a negative and a negative.1750

I have factors of 16, which are 1 and 16, 2 and 8, and 4 and 4; and I need factors of 16 that will add up to -8.1755

And I can see that this one is correct, (x - 4) (x - 4), because that is going to give me a middle term of -8x.1767

OK, this second set of factors is going to be x and x, and everything is positive.1782

And factors of 4 that would add up to 4 would be 2 and 2: so, x2 + 2x + 2x (that is going to give me 4x) + 4.1792

Finally, I have a negative here; so I am going to do a plus here and a negative here.1806

I want factors of 8 that add up to -2: well, 1 and 8 are too far apart, so I have 2 and 4.1814

And I want it to be a -2, so I am going to make the 4 negative.1824

This is going to give me a 2 here and a 4 here.1828

OK, I could actually rewrite this, also, as (x - 4)2; and I am going to rewrite this as (x + 2)2.1832

So, when I look for my LCM, I am going to look for each factor that I have.1844

And the first one is (x - 4), and the highest power I have it to is 2; I have an (x - 4) here, but this is really only to the first power.1849

So, I am going to write this as (x - 4)2.1858

I took care of that factor that is right here, also.1865

Now, I have another unique factor of (x + 2).1869

The highest power in any one polynomial that I find it in is squared; I have an (x + 2) here,1874

but it is to a lower power, so I am not going to worry about that; it is covered under this.1879

The least common multiple of these three polynomials is (x - 4)2 (x + 2)2.1884

Factor out each polynomial, and then take the product of their factors and use the power1891

that is the highest power that any factor is present in, in one of the polynomials.1897

Here is addition: adding rational expressions with different denominators.1905

The first step is to get a common denominator.1910

To achieve that, I am going to factor the denominators.1915

There is a negative here, so this is plus; this is minus.1925

I need factors of 6 that add up to -1; I am going to look at these two, and if I make the 3 negative, then I am going to get the correct middle term.1930

So, I am going to put the 3 here and the 2 here.1943

OK, so I factored out that first denominator; let's look at the second one.1946

Leave the numerator alone for now, and concentrate on the denominator.1952

I can see, pretty quickly, that I have a common factor of 4; so that will become x, and this will become minus 3.1956

So, this is what I want to add; and I need to find the LCD, so I am going to look at all of these factors that I have in the denominator and find their product.1965

I have (x + 2), and that only appears once, so I just leave it as the first power.1978

I have (x - 3) here and here, and the highest number of times it appears is once and once; so it is (x - 3).1984

Over here, I also have a 4; so the LCD...I am going to rewrite this with a 4 in front...is going to be 4(x + 2) (x - 3).1994

Now, I need to rewrite these as equivalent fractions with this denominator.2007

I look at the denominator, and I see what I am lacking.2014

In this first denominator, I have (x + 2) (x - 3), but I am lacking a 4.2018

So, I am going to multiply both the numerator and the denominator by 4 to form an equivalent fraction.2025

This is going to give me 4(2x - 3)/(x + 2)(x - 3).2036

Here, I am going to end up with...let's work with this one right down here...(3x2 - 2x)/(4(x - 3)).2048

What is lacking from the denominator? (x + 2)2061

I have my 4; I have my (x - 3); I am lacking an (x + 2).2065

So, I am going to multiply both the numerator and the denominator by that.2070

This is going to give me (x + 2)(3x2 - 2x), all over...oops, I need a 4 down there, as well;2078

this 4 should be over there...times 4, times (x - 3)(x + 2).2093

OK, I have a common denominator--it is a slightly different order that I wrote it in, but I still have a 4, an (x + 2), and an (x - 3).2100

Now, I can add these: I am going to go ahead--I have my equivalent fractions, and I am going to add these.2106

4(2x - 3) divided by 4(x + 2) (x - 3), plus (x + 2)(3x2 - 2x) over this common denominator,2113

4...I am going to write this in the same order as I wrote this one...(x + 2) (x - 3).2133

Now, once you have a common denominator, all you need to do is add the numerators and put these over the common denominator.2139

And the common denominator here is 4(x + 2) (x - 3).2160

Here, this is all factored out; I need to add this and then see what I have an if there are common factors.2170

I need to go ahead and multiply this all out.2176

In the numerator, this is 4 times 2x (that is 8x) minus 12; here I have x times 3x2, gives me 3x3.2179

x times -2x gives me -2x2; I took care of this and this using the distributive property.2197

2 times 3x2 is 6x2; 2 times -2x is -4x.2212

This is all, again, over that common denominator.2220

Now, I can do a little more simplifying, because I can add like terms.2226

My denominator is taken care of; let's look at this numerator.2231

Starting with the largest power: I only have one x3 term, so that is 3x3.2237

For x2 terms, I have 6x2 - 2x2; that is going to give me 4x2.2243

For x terms, I have...let's see...8x - 4x; that is going to give me 4x; for constants, I have -12.2249

Now, what I am left with is this rational expression.2266

And in order to simplify this, the way that you have to go about it is to use synthetic division.2269

And there actually turn out to be no common factors, so I am going to leave it as it is.2276

But you could check it by synthetic division; and the way to proceed would be to use synthetic division, and to divide this polynomial by (x + 2).2282

And remember from the remainder theorem: if the remainder is 0, then (x + 2) is a common factor, and you would be able to cancel that out.2292

I know 4 is not a common factor; I could also use synthetic division to determine that (x - 3) is not a factor of this.2299

But if you did have common factors, then the final step would be to cancel those out.2306

So, this was a pretty complicated problem; but proceeding as usual, factor the denominators, finding the least common denominator.2312

Then convert to equivalent fractions by multiplying the numerator and denominator by whatever was lacking from the denominator to form the LCD.2321

And this was 4 before, times this first rational expression.2337

Down here, I had to multiply the second one times (x + 2), divided by (x + 2).2343

I ended up with these two, with a common denominator; I rewrote them up here, and then I just added the numerators and did simplification.2349

OK, in this example, we are going to be subtracting rational expressions with different denominators.2358

The first thing to do is factor out the denominators to find the LCD.2365

Factoring the first one: (x - 7) divided by...I am going to have an x here and an x here...2369

Now, this is a negative sign; so I am going to have + and -.2376

Factors of 12 are 1 and 12, 2 and 6, 3 and 4; factors of 12 that would add up to -4 would be these two, 2 - 6.2381

So, I am going to put my 2 by the positive sign, and the 6 by the negative sign.2397

And I am going to leave some space here for when I convert these to equivalent fractions.2404

I am going to put my negative sign right here; and this is going to give me 2x + 3, and I am going to go ahead and factor this.2409

This one is a little bit more complicated to factor, because the leading coefficient is not 1.2417

So, I am going to get 2x here and an x here.2422

Now, I have a negative sign here, but I have a positive sign in front of the constant.2426

And that tells me that I am working with a negative and a negative, which will give me a positive here and a negative here.2431

This is more complicated, because I have to take into account this 2.2440

So, let's think about factors of 18, first of all, which are 1 and 18, 2 and 9, and 3 and 6.2445

Now, when I am working with a leading coefficient other than 1, I like to start out with smaller numbers,2456

because the one that is being multiplied with the 2 is going to become large.2461

So, let's start out with 3 and 6 and look at different combinations.2466

If I have 2x (let's put the 3 first) - 3, times (x - 6), that is going to give me 2x2;2471

the outer terms give -12x; the inner terms give -3x; and the last terms give 18.2482

Since -12x and -3x add up to -15x, this is the correct factorization.2490

OK, now thinking about the LCD (least common denominator): I look at the factors I have, and I have an (x + 2).2499

And it is only present once, so it is just a power of 1.2510

I also have (x - 6), and I have one here, and it is only present once here; so again, I am just going to represent that once.2516

And so, this one is taken care of; over here, I also have (2x - 3).2525

Therefore, the LCD is going to be (x + 2) (x - 6) (2x - 3).2530

Now, I need to convert these to equivalent fractions.2536

To do that, I am going to multiply the numerator and the denominator by what is lacking from the denominator.2539

Here, the factor that is missing is 2x - 3, so I need to multiply both the numerator and the denominator by that.2547

Over here, I have the (2x - 3); I have (x - 6); but I need to multiply both the numerator and the denominator by (x + 2).2559

Now, when I multiply these out, I am going to end up with a common denominator.2570

I am going to take care of that multiplication: (x - 7) times (2x - 3), over (x + 2) times (x - 6) times (2x - 3);2574

minus this entire second fraction, which is going to be (2x + 3) times (x + 2), divided by the common denominator.2595

I am going to go ahead and write this in the same order as this one: (x + 2) first; (x - 6); and then (2x - 3).2611

OK, I am just checking to make sure that I have everything accounted for.2629

Now that I have a common denominator, I can subtract; so this is going to become (x - 7) times (2x - 3).2632

And I need to be careful with the signs; it is going to be subtracting, so this whole thing is going to be the opposite signs...over the common denominator.2644

OK, what is left is to simplify: this is already factored out, but I need to multiply this out,2666

add together the like terms, and then factor and see if I can simplify.2673

So, starting out right here, this is x times 2x; that gives me 2x2; x times -3...that is -3x;2680

here I get -14x (that is -7 times 2x); and then -7 times -3 is 21.2699

OK, minus what is in here: so, 2x times x is 2x2; 2x times 2 is 4x; now, the second term:2711

3 times x is 3x; 3 times 2 is 6; all over the common denominator, (x + 2) (x - 6) (2x - 3).2732

The next thing to do is take care of these signs: this equals 2x2...and do some simplifying.2749

-3x and -14x is -17x, plus 21; here I am going to have a negative; that gives me -2x2.2757

Inside here, I have 4x and 3x, so that is 7x; but I need to take the negative of that--the opposite; this is actually -7x.2772

For the constant I have 6, and the opposite of that is -6, all over the common denominator.2780

Now, I have some like terms that can be combined.2790

So, I am going to go down here; and this gives me 2x2 - 2x2; these cancel, so the x2 terms are gone.2793

I have -17x and -7x combined, to give -24x; that leaves me with the constants: 21 - 6 is 15...over the common denominator.2806

OK, looking at this, I can see that I don't have any common factors, so I can't simplify.2834

You could pull a 3 from up here--you could factor out a 3; but that is not going to leave you with any common factors.2846

There is no (x + 2), (x - 6), or (2x - 3) that is going to be left behind; so I can just leave this as it is.2856

This was a pretty complex problem--pretty lengthy.2863

Since it is subtraction, you have to be careful with the signs.2867

Again, you are factoring the denominators of both, finding the LCD right here, then converting to equivalent fractions.2869

This first fraction was lacking the (2x - 3) in the denominator; so I multiplied both by that.2878

The second fraction needed an (x + 2) multiplied by both the numerator and the denominator.2886

Once I did that, then it was a matter of subtracting, and I had the same denominator.2892

I had to do some multiplying, combining, and simplifying to end up with the final answer that I have here.2898

OK, here I have a complex fraction; and not only is it a complex fraction,2906

but the rational expressions that we see are being added or subtracted in the numerator and the denominator.2911

Again, the way to handle this is to handle the numerator and the denominator separately.2917

So, first the numerator: I want to subtract this.2922

The common denominator: well, my LCD is going to be x2y3.2934

I look at what I have and what is lacking: well, what is lacking from this denominator, x2, is a y3.2943

So, I am going to multiply both the numerator and the denominator by y3, minus 1/y3.2952

What is lacking here from the denominator is the x2, so I am multiplying both the numerator and the denominator by x2.2963

This is going to give me y3 right here, over x2y3, minus x2 over x2y3.2972

Since I now have a common denominator, I can then subtract.2988

So, it is y3 - x2, over this common denominator.2994

This is my numerator; so I am going to go over here and write this new numerator that I have.3000

And this is much easier to work with, because now I just have a fraction.3006

I have a rational expression; I don't have two rational expressions and subtraction.3010

The denominator: that was the numerator--let's now work with the denominator.3016

I am adding, and I am being asked to add (y/x3) + (x/y2).3027

The LCD is x3y2; to convert this, I am going to have to multiply this fraction3034

by y2/y2, because that is what is lacking.3051

And I am going to add that to x/y2; and what is lacking from this denominator is x3.3065

I multiply this; this is x3 over x3.3072

This gives me y times y2 (is y3), over the common denominator, x3y2,3077

plus x times x3 (is going to give me x4), over x3y2.3086

Since these now have a common denominator, I am going to add y3 + x4, over x3y2.3102

OK, I handled this as two different problems: a subtraction problem up here to get the numerator,3112

and an addition problem adding rational expressions down here, to find the denominator,3118

which is y3 + x4, over x3y2.3123

Once I am to this point, I just use my usual rules for dividing rational expressions.3134

So remember: we are going to rewrite this as a division problem, because this fraction bar is just telling me to divide.3140

This is going to give me...I am going to rewrite this down here as (y3 - x2), divided by x3y2.3147

This entire rational expression is being divided by this one: y3 + x4 divided by x3y2.3158

Dividing one rational expression by another is simply multiplying the first by the inverse of the second.3171

So, I am going to rewrite this as x3y2, divided by y3 + x4.3184

Now, multiplication: the next step is always to simplify.3194

So, let's look for common factors: I have x3 here and x2 down here--get rid of the x2;3198

this becomes x, because I took out that factor of x2.3208

I have a y2 here and a y3 here; that cancels out, and this just becomes y, because I took out a y2.3213

Now, let's see what I have left and multiply that.3222

I have an x, times this whole thing, which is y3 - x2, divided by...3224

I have a y left here, and I have y3 + x4.3239

And now, I have simplified it as far as I can.3246

And this took many steps; you have to be careful and make sure that you keep track of everything.3252

But start out by simplifying the numerator, by subtracting to get this numerator.3257

Simplify the denominator by adding to get this for the denominator.3264

Then, treat this as a regular complex fraction, where we are going to take this numerator and divide by the denominator.3272

And we handle that by taking the first rational expression and multiplying by the reciprocal of the second.3283

I found common factors; I canceled those out; and this is what I ended up with.3289

And it cannot be simplified any more.3295

That concludes this lesson on adding and subtracting rational expressions.3298

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