  Eric Smith

Multiply & Divide Rational Expressions

Slide Duration:

Section 1: Properties of Real Numbers
Basic Types of Numbers

30m 41s

Intro
0:00
Objectives
0:07
Basic Types of Numbers
0:36
Natural Numbers
1:02
Whole Numbers
1:29
Integers
2:04
Rational Numbers
2:38
Irrational Numbers
5:06
Imaginary Numbers
6:48
Basic Types of Numbers Cont.
8:09
The Big Picture
8:10
Real vs. Imaginary Numbers
8:30
Rational vs. Irrational Numbers
8:48
Basic Types of Numbers Cont.
10:55
Number Line
11:06
Absolute Value
11:44
Inequalities
12:39
Example 1
13:16
Example 2
17:30
Example 3
21:56
Example 4
24:27
Example 5
27:48
Operations on Numbers

19m 26s

Intro
0:00
Objectives
0:06
Operations on Numbers
0:25
0:53
Subtraction
1:33
Multiplication & Division
2:19
Exponents
3:24
Bases
4:04
Square Roots
4:59
Principle Square Roots
5:09
Perfect Squares
6:32
Simplifying and Combining Roots
6:52
Example 1
8:16
Example 2
12:30
Example 3
14:02
Example 4
16:27
Order of Operations

12m 6s

Intro
0:00
Objectives
0:06
The Order of Operations
0:25
Work Inside Parentheses
0:42
Simplify Exponents
0:52
Multiplication & Division from Left to Right
0:57
Addition & Subtraction from Left to Right
1:11
Remember PEMDAS
1:21
The Order of Operations Cont.
2:27
Example
2:43
Example 1
3:55
Example 2
5:36
Example 3
7:35
Example 4
8:56
Properties of Real Numbers

18m 52s

Intro
0:00
Objectives
0:07
The Properties of Real Numbers
0:23
Commutative Property of Addition and Multiplication
0:44
Associative Property of Addition and Multiplication
1:50
Distributive Property of Multiplication Over Addition
3:20
Division Property of Zero
4:46
Division Property of One
5:23
Multiplication Property of Zero
5:56
Multiplication Property of One
6:17
6:29
Why Are These Properties Important?
6:53
Example 1
9:16
Example 2
13:04
Example 3
14:30
Example 4
16:57
Section 2: Linear Equations
The Vocabulary of Linear Equations

12m 22s

Intro
0:00
Objectives
0:09
The Vocabulary of Linear Equations
0:44
Variables
0:52
Terms
1:09
Coefficients
1:40
Like Terms
2:18
Examples of Like Terms
2:37
Expressions
4:01
Equations
4:26
Linear Equations
5:04
Solutions
5:55
Example 1
6:16
Example 2
7:16
Example 3
8:45
Example 4
10:20
Solving Linear Equations in One Variable

28m 52s

Intro
0:00
Objectives
0:08
Solving Linear Equations in One Variable
0:34
Conditional Cases
0:51
Identity Cases
1:09
1:30
Solving Linear Equations in One Variable Cont.
2:00
2:10
Multiplication Property of Equality
2:43
Steps to Solve Linear Equations
3:14
Example 1
4:22
Example 2
8:21
Example 3
12:32
Example 4
14:19
Example 5
17:25
Example 6
22:17
Solving Formulas

12m 2s

Intro
0:00
Objectives
0:06
Solving Formulas
0:18
Formulas
0:26
Use the Same Properties as Solving Linear Equations
1:36
1:55
Multiplication Property of Equality
1:58
Steps to Solve Formulas
2:43
Example 1
3:56
Example 2
6:09
Example 3
8:39
Applications of Linear Equations

28m 41s

Intro
0:00
Objectives
0:10
Applications of Linear Equations
0:43
The Six-Step Method to Solving Word Problems
0:55
Common Terms
3:12
Example 1
5:03
Example 2
9:40
Example 3
13:48
Example 4
17:58
Example 5
23:28
Applications of Linear Equations, Motion & Mixtures

24m 26s

Intro
0:00
Objectives
0:21
Motion and Mixtures
0:46
Motion Problems: Distance, Rate, and Time
1:06
Mixture Problems: Amount, Percent, and Total
1:27
The Table Method
1:58
The Beaker Method
3:38
Example 1
5:05
Example 2
9:44
Example 3
14:20
Example 4
19:13
Section 3: Graphing
Rectangular Coordinate System

22m 55s

Intro
0:00
Objectives
0:11
The Rectangular Coordinate System
0:39
The Cartesian Coordinate System
0:40
X-Axis
0:54
Y-Axis
1:04
Origin
1:11
1:26
Ordered Pairs
2:10
Example 1
2:55
The Rectangular Coordinate System Cont.
6:09
X-Intercept
6:45
Y-Intercept
6:55
Relation of X-Values and Y-Values
7:30
Example 2
11:03
Example 3
12:13
Example 4
14:10
Example 5
18:38
Slope & Graphing

27m 58s

Intro
0:00
Objectives
0:11
Slope and Graphing
0:48
Standard Form
1:14
Example 1
2:24
Slope and Graphing Cont.
4:58
Slope, m
5:07
Slope is Rise over Run
6:11
Don't Mix Up the Coordinates
8:20
Example 2
9:39
Slope and Graphing Cont.
14:26
Slope-Intercept Form
14:34
Example 3
16:55
Example 4
18:00
Slope and Graphing Cont.
19:00
Rewriting an Equation in Slope-Intercept Form
19:39
Rewriting an Equation in Standard Form
20:09
Slopes of Vertical & Horizontal Lines
20:56
Example 5
22:49
Example 6
24:09
Example 7
25:59
Example 8
26:57
Linear Equations in Two Variables

20m 36s

Intro
0:00
Objectives
0:13
Linear Equations in Two Variables
0:36
Point-Slope Form
1:07
Substitute in the Point and the Slope
2:21
Parallel Lines: Two Lines with the Same Slope
4:05
Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
4:39
Perpendicular Lines: Product of Slopes is -1
5:24
Example 1
6:02
Example 2
7:50
Example 3
10:49
Example 4
13:26
Example 5
15:30
Example 6
17:43
Section 4: Functions
Introduction to Functions

21m 24s

Intro
0:00
Objectives
0:07
Introduction to Functions
0:58
Relations
1:03
Functions
1:37
Independent Variables
2:00
Dependent Variables
2:11
Function Notation
2:21
Function
3:43
Input and Output
3:53
Introduction to Functions Cont.
4:45
Domain
4:46
Range
4:55
Functions Represented by a Diagram
6:41
Natural Domain
9:11
Evaluating Functions
12:02
Example 1
13:13
Example 2
15:03
Example 3
16:18
Example 4
19:54
Graphing Functions

16m 12s

Intro
0:00
Objectives
0:09
Graphing Functions
0:54
Using Slope-Intercept Form
1:56
Vertical Line Test
2:58
Determining the Domain
4:20
Determining the Range
5:43
Example 1
6:06
Example 2
7:18
Example 3
8:31
Example 4
11:04
Section 5: Systems of Linear Equations
Systems of Linear Equations

25m 54s

Intro
0:00
Objectives
0:13
Systems of Linear Equations
0:46
System of Equations
0:51
System of Linear Equations
1:15
Solutions
1:35
Points as Solutions
1:53
Finding Solutions Graphically
5:13
Example 1
6:37
Example 2
12:07
Systems of Linear Equations Cont.
17:01
One Solution, No Solution, or Infinite Solutions
17:10
Example 3
18:31
Example 4
22:37
Solving a System Using Substitution

20m 1s

Intro
0:00
Objectives
0:09
Solving a System Using Substitution
0:32
Substitution Method
1:24
Substitution Example
2:35
One Solution, No Solution, or Infinite Solutions
7:50
Example 1
9:45
Example 2
12:48
Example 3
15:01
Example 4
17:30
Solving a System Using Elimination

19m 40s

Intro
0:00
Objectives
0:09
Solving a System Using Elimination
0:27
Elimination Method
0:42
Elimination Example
2:01
One Solution, No Solution, or Infinite Solutions
7:05
Example 1
8:53
Example 2
11:46
Example 3
15:37
Example 4
17:45
Applications of Systems of Equations

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
Section 6: Inequalities
Solving Linear Inequalities in One Variable

17m 13s

Intro
0:00
Objectives
0:08
Solving Linear Inequalities in One Variable
0:37
Inequality Expressions
0:46
Linear Inequality Solution Notations
3:40
Inequalities
3:51
Interval Notation
4:04
Number Lines
4:43
Set Builder Notation
5:24
Use Same Techniques as Solving Equations
6:59
'Flip' the Sign when Multiplying or Dividing by a Negative Number
7:12
'Flip' Example
7:50
Example 1
8:54
Example 2
11:40
Example 3
14:01
Compound Inequalities

16m 13s

Intro
0:00
Objectives
0:07
Compound Inequalities
0:37
'And' vs. 'Or'
0:44
'And'
3:24
'Or'
3:35
'And' Symbol, or Intersection
3:51
'Or' Symbol, or Union
4:13
Inequalities
4:41
Example 1
6:22
Example 2
9:30
Example 3
11:27
Example 4
13:49
Solving Equations with Absolute Values

14m 12s

Intro
0:00
Objectives
0:08
Solve Equations with Absolute Values
0:18
Solve Equations with Absolute Values Cont.
1:11
Steps to Solving Equations with Absolute Values
2:21
Example 1
3:23
Example 2
6:34
Example 3
10:12
Inequalities with Absolute Values

17m 7s

Intro
0:00
Objectives
0:07
Inequalities with Absolute Values
0:23
Recall…
2:08
Example 1
3:39
Example 2
6:06
Example 3
8:14
Example 4
10:29
Example 5
13:29
Graphing Inequalities in Two Variables

15m 33s

Intro
0:00
Objectives
0:07
Graphing Inequalities in Two Variables
0:32
Split Graph into Two Regions
1:53
Graphing Inequalities
5:44
Test Points
6:20
Example 1
7:11
Example 2
10:17
Example 3
13:06
Systems of Inequalities

21m 13s

Intro
0:00
Objectives
0:08
Systems of Inequalities
0:24
Test Points
1:10
Steps to Solve Systems of Inequalities
1:25
Example 1
2:23
Example 2
7:28
Example 3
12:51
Section 7: Polynomials
Integer Exponents

44m 51s

Intro
0:00
Objectives
0:09
Integer Exponents
0:42
Exponents 'Package' Multiplication
1:25
Example 1
2:00
Example 2
3:13
Integer Exponents Cont.
4:50
Product Rule for Exponents
4:51
Example 3
7:16
Example 4
10:15
Integer Exponents Cont.
13:13
Power Rule for Exponents
13:14
Power Rule with Multiplication and Division
15:33
Example 5
16:18
Integer Exponents Cont.
20:04
Example 6
20:41
Integer Exponents Cont.
25:52
Zero Exponent Rule
25:53
Quotient Rule
28:24
Negative Exponents
30:14
Negative Exponent Rule
32:27
Example 7
34:05
Example 8
36:15
Example 9
39:33
Example 10
43:16

18m 33s

Intro
0:00
Objectives
0:07
0:25
Terms
0:33
Coefficients
0:51
1:13
Like Terms
1:29
Polynomials
2:21
Monomials, Binomials, Trinomials, and Polynomials
5:41
Degrees
7:00
Evaluating Polynomials
8:12
9:25
Example 1
11:48
Example 2
13:00
Example 3
14:41
Example 4
16:15
Multiplying Polynomials

25m 7s

Intro
0:00
Objectives
0:06
Multiplying Polynomials
0:41
Distributive Property
1:00
Example 1
2:49
Multiplying Polynomials Cont.
8:22
Organize Terms with a Table
8:23
Example 2
13:40
Multiplying Polynomials Cont.
16:33
Multiplying Binomials with FOIL
16:48
Example 3
18:49
Example 4
20:04
Example 5
21:42
Dividing Polynomials

44m 56s

Intro
0:00
Objectives
0:07
Dividing Polynomials
0:29
Dividing Polynomials by Monomials
2:10
Dividing Polynomials by Polynomials
2:59
Dividing Numbers
4:09
Dividing Polynomials Example
8:39
Example 1
12:35
Example 2
14:40
Example 3
16:45
Example 4
21:13
Example 5
24:33
Example 6
29:02
Dividing Polynomials with Synthetic Division Method
33:36
Example 7
38:43
Example 8
42:24
Section 8: Factoring Polynomials
Greatest Common Factor & Factor by Grouping

28m 27s

Intro
0:00
Objectives
0:09
Greatest Common Factor
0:31
Factoring
0:40
Greatest Common Factor (GCF)
1:48
GCF for Polynomials
3:28
Factoring Polynomials
6:45
Prime
8:21
Example 1
9:14
Factor by Grouping
14:30
Steps to Factor by Grouping
17:03
Example 2
17:43
Example 3
19:20
Example 4
20:41
Example 5
22:29
Example 6
26:11
Factoring Trinomials

21m 44s

Intro
0:00
Objectives
0:06
Factoring Trinomials
0:25
Recall FOIL
0:26
Factor a Trinomial by Reversing FOIL
1:52
Tips when Using Reverse FOIL
5:31
Example 1
7:04
Example 2
9:09
Example 3
11:15
Example 4
13:41
Factoring Trinomials Cont.
15:50
Example 5
18:42
Factoring Trinomials Using the AC Method

30m 9s

Intro
0:00
Objectives
0:08
Factoring Trinomials Using the AC Method
0:27
Factoring when Leading Term has Coefficient Other Than 1
1:07
Reversing FOIL
1:18
Example 1
1:46
Example 2
4:28
Factoring Trinomials Using the AC Method Cont.
7:45
The AC Method
8:03
Steps to Using the AC Method
8:19
Tips on Using the AC Method
9:29
Example 3
10:45
Example 4
16:50
Example 5
21:08
Example 6
24:58
Special Factoring Techniques

30m 14s

Intro
0:00
Objectives
0:07
Special Factoring Techniques
0:26
Difference of Squares
1:46
Perfect Square Trinomials
2:38
No Sum of Squares
3:32
Special Factoring Techniques Cont.
4:03
Difference of Squares Example
4:04
Perfect Square Trinomials Example
5:29
Example 1
7:31
Example 2
9:59
Example 3
11:47
Example 4
15:09
Special Factoring Techniques Cont.
19:07
Sum of Cubes and Difference of Cubes
19:08
Example 5
23:13
Example 6
26:12

23m 38s

Intro
0:00
Objectives
0:08
0:19
0:20
Zero Factor Property
1:39
Zero Factor Property Example
2:34
Example 1
4:00
Solving Quadratic Equations by Factoring Cont.
5:54
Example 2
7:28
Example 3
11:09
Example 4
14:22
Solving Quadratic Equations by Factoring Cont.
18:17
Higher Degree Polynomial Equations
18:18
Example 5
20:22

29m 27s

Intro
0:00
Objectives
0:12
0:29
Linear Factors
0:38
1:22
Principle of Square Roots
3:36
Completing the Square
4:50
Steps for Using Completing the Square
5:15
Completing the Square Works on All Quadratic Equations
6:41
7:28
Discriminants
8:25
10:11
Example 1
11:54
Example 2
13:03
Example 3
16:30
Example 4
21:29
Example 5
25:07

16m 47s

Intro
0:00
Objectives
0:08
0:24
Using a Substitution
0:53
U-Substitution
1:26
Example 1
2:07
Example 2
5:36
Example 3
8:31
Example 4
11:14

29m 4s

Intro
0:00
Objectives
0:09
0:35
Squared Variable
0:40
Principle of Square Roots
0:51
Example 1
1:09
Example 2
2:04
3:34
Example 3
4:42
Example 4
13:33
Example 5
20:50

26m 53s

Intro
0:00
Objectives
0:06
0:39
Axis of Symmetry
1:46
Vertex
2:12
Transformations
2:57
3:23
Example 1
5:06
Example 2
6:02
Example 3
9:07
11:26
Completing the Square
12:02
Vertex Shortcut
12:16
Example 4
13:49
Example 5
17:25
Example 6
20:07
Example 7
23:43
Polynomial Inequalities

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
Section 10: Rational Equations
Multiply & Divide Rational Expressions

26m 41s

Intro
0:00
Objectives
0:09
Multiply and Divide Rational Expressions
0:44
Rational Numbers
0:55
Dividing by Zero
1:45
Canceling Extra Factors
2:43
Negative Signs in Fractions
4:52
Multiplying Fractions
6:26
Dividing Fractions
7:17
Example 1
8:04
Example 2
14:01
Example 3
16:23
Example 4
18:56
Example 5
22:43

20m 24s

Intro
0:00
Objectives
0:07
0:41
Common Denominators
0:52
Common Denominator Examples
1:14
Steps to Adding and Subtracting Rational Expressions
2:39
Example 1
3:34
Example 2
5:27
Adding and Subtracting Rational Expressions Cont.
6:57
Least Common Denominators
6:58
Transitioning from Fractions to Rational Expressions
9:08
Identifying Least Common Denominators for Rational Expressions
9:56
10:41
Example 3
11:19
Example 4
12:36
Example 5
15:08
Example 6
16:46
Complex Fractions

18m 23s

Intro
0:00
Objectives
0:09
Complex Fractions
0:37
Dividing to Simplify Complex Fractions
1:10
Example 1
2:03
Example 2
3:58
Complex Fractions Cont.
9:15
Using the Least Common Denominator to Simplify Complex Fractions
9:16
10:07
Example 3
10:42
Example 4
14:28
Solving Rational Equations

16m 24s

Intro
0:00
Objectives
0:07
Solving Rational Equations
0:23
Isolate the Specified Variable
1:23
Example 1
1:58
Example 2
5:00
Example 3
8:23
Example 4
13:25
Rational Inequalities

18m 54s

Intro
0:00
Objectives
0:06
Rational Inequalities
0:18
Testing Intervals for Rational Inequalities
0:38
Steps to Solving Rational Inequalities
1:05
Tips to Solving Rational Inequalities
2:27
Example 1
3:33
Example 2
12:21
Applications of Rational Expressions

20m 20s

Intro
0:00
Objectives
0:07
Applications of Rational Expressions
0:27
Work Problems
1:05
Example 1
2:58
Example 2
6:45
Example 3
13:17
Example 4
16:37
Variation & Proportion

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
Rational Exponents

14m 32s

Intro
0:00
Objectives
0:07
Rational Exponents
0:32
Power on Top, Root on Bottom
1:05
Example 1
1:37
Rational Exponents Cont.
4:04
Using Rules from Exponents for Radicals as Exponents
4:05
Combining Terms Under a Single Root
4:50
Example 2
5:21
Example 3
7:39
Example 4
11:23
Example 5
13:14
Simplify Rational Exponents

15m 12s

Intro
0:00
Objectives
0:07
Simplify Rational Exponents
0:25
0:26
Product Rule to Simplify Square Roots
1:11
1:42
Applications of Product and Quotient Rules
2:17
Higher Roots
2:48
Example 1
3:39
Example 2
6:35
Example 3
8:41
Example 4
11:09

17m 22s

Intro
0:00
Objectives
0:07
0:33
Like Terms
1:29
Bases and Exponents May be Different
2:02
Bases and Powers Must be Same when Adding and Subtracting
2:42
3:55
Example 1
4:47
Example 2
6:00
7:10
Simplify the Bases to Look the Same
7:25
Example 3
8:23
Example 4
11:45
Example 5
15:10

19m 24s

Intro
0:00
Objectives
0:08
0:25
0:26
1:11
Don’t Distribute Powers
2:54
4:25
Rationalizing Denominators
6:40
Example 1
7:22
Example 2
8:32
9:23
Rationalizing Denominators with Higher Roots
9:25
Example 3
10:51
Example 4
11:53
13:13
Rationalizing Denominators with Conjugates
13:14
Example 5
15:52
Example 6
17:25

15m 5s

Intro
0:00
Objectives
0:07
0:17
0:18
Isolate the Roots and Raise to Power
0:34
Example 1
1:13
Example 2
3:09
7:04
7:05
Example 3
7:54
Example 4
13:07
Complex Numbers

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33
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• ## Related Books 1 answerLast reply by: Ming ZhangSat May 2, 2020 4:47 PMPost by Rose A on November 9, 2019Absolutely amazing! I learned this in my advanced grade 10math and needed more revision before my exam!! this is doing the jobbthanks! 1 answer Last reply by: Professor Eric SmithThu Sep 5, 2019 11:30 AMPost by chengpingru on September 4, 2019On  example 1 question 2, you said that (w^2 - 5)/(5-w^2) = (-1)(-w^2+5)/(5-w^2) = (-1)(5-w^2)/(5-w^2)But when you canceled the numerator and the denominator out, you changed 5-w^2 into 5+w^2 to match the denominator, which canceled them out. I am confused about how/why you changed a previous answer that is correct into something incorrect. Perhaps a mistake? 1 answer Last reply by: Professor Eric SmithTue Aug 20, 2019 5:04 PMPost by Eva Wu on August 20, 2019In the Example 5 slide, it said Example 4 not Example 5 1 answer Last reply by: Professor Eric SmithTue Aug 20, 2019 5:03 PMPost by Eva Wu on August 20, 2019Why where there Varibles in the denominator for example 3 when Polynomials cant have variables in the denominator? (There may be more than one example where this happens) 0 answersPost by GERARDO MORALES on October 3, 2013If theres an option to fast forward your lecture? 1 answer Last reply by: Professor Eric SmithThu Oct 3, 2013 5:27 PMPost by GERARDO MORALES on October 3, 2013If theres an option to fast forward your lecture?

### Multiply & Divide Rational Expressions

• A rational expression involves one polynomial being divided by another polynomial
• To simplify a rational expression we want to cancel out common factors in the numerator and denominator. This is similar to simplifying fractions.
• To multiply rational expressions, we multiply across the top and bottom of the fractions.
• To divide rational expressions, flip the second rational expression and multiply.
• It is often best to factor the polynomials first before combining using multiplication. This makes the simplification process a bit easier.

### Multiply & Divide Rational Expressions

Multiply:
[(40a2b3c5)/(20a3b5c4)] ×[(15a4b3c2)/(24a2b5c3)]
• [(1c2)/(4b2c2)] ×[3a/(6b2)]
• [1/(4b2)] ×[a/(2b2)]
• [a/(4b2 ×2b2)]
[a/(8b4)]
Multiply:
[(18u4v3w5x8)/(26v4w6x2)] ×[(14v5w2x4)/(54u3v6w7)]
• [(1ux8)/(13w4)] ×[(7vx2)/(3v3w2)]
• [(ux8)/(13w4)] ×[(7x2)/(3v2w5)]
[(7ux10)/(39v2w9)]
Multiply:
[(42c3d2e4)/(60c5d6e2)] ×[(45c6e2f3)/(35d4ef2)]
• [(6c3)/(4d6)] ×[(3cf3)/(5d2ef2)]
• [(3c3)/(2d6)] ×[3cf/(5d2e)]
[(9c4f)/(10d8e)]
Multiply:
[(6y + 36)/(4y − 14)] ×[(16y + 30)/(2y + 12)]
• [(6( y + 6 ))/(2( 2y − 7 ))] ×[(2( 8y + 15 ))/(2( y + 6 ))]
• [3/(2y − 7)] ×[(8y + 15)/1]
[(3( 8y + 15 ))/(2y − 7)]
Multiply:
[(24n − 36)/(10n − 35)] ×[(16n + 48)/(18n − 27)]
• [(12( 2n − 3 ))/(5( 2n − 7 ))] ×[(16( n + 3 ))/(9( 2n − 3 ))]
• [12/(5( 2n − 7 ))] ×[(16( n + 3 ))/9]
[4/(5( 2n − 7 ))] ×[(16( n + 3 ))/3]
Multiply:
[(16b + 44)/(27b − 45)] ×[(15b − 25)/(16b − 72)]
• [(4( 4b + 11 ))/(9( 3b − 5 ))] ×[(5( 3b − 5 ))/(8( 2b − 9 ))]
[(( 4b + 11 ))/9] ×[5/(( 2b − 9 ))]
Multiply:
[(9x + 12)/(4x + 20)] ×[(x2 + 3x − 10)/(3x2 + x − 4)]
• [(3( 3x + 4 ))/(4( x + 5 ))] ×[(( x + 5 )( x − 2 ))/(( 3x + 4 )( x − 1 ))]
• [3/4] ×[(x − 2)/(x − 1)]
[(3( x − 2 ))/(4( x − 1 ))]
Multiply:
[(6x − 21)/(6x − 24)] ×[(5x2 + 25x + 20)/(2x2 + x − 28)]
• [(3( 2x − 7 ))/(6( x + 4 ))] ×[(5( x2 + 5x + 4 ))/(2x2 + x − 28)]
• [(3( 2x − 7 ))/(6( x + 4 ))] ×[(5( x + 4 )( x + 1 ))/(( 2x − 7 )( x + 4 ))]
• [3/6] ×[(5( x + 1 ))/(( x + 4 ))]
• [1/2] ×[(5( x + 1 ))/(( x + 4 ))]
[(5( x + 1 ))/(2( x + 4 ))]
Multiply:
[(x2 − 12x + 32)/(4x2 − 16x − 20)] ×[(2x2 − 50)/(2x2 − 15x − 8)]
• [(( x − 4 )( x − 8 ))/(4( x2 − 4x − 5 ))] ×[(2( x2 − 25 ))/(2x2 − 15x − 8)]
• [(( x − 4 )( x − 8 ))/(4( x + 1 )( x − 5 ))] ×[(2( x + 5 )( x − 5 ))/(( 2x + 1 )( x − 8 ))]
• [(x − 4)/(4( x + 1 ))] ×[(2( x + 5 ))/(2x + 1)]
• [(x − 4)/(2( x + 1 ))] ×[(x + 5)/(2x + 1)]
[(( x − 4 )( x + 5 ))/(2( x + 1 )( 2x + 1 ))]
Multiply:
[(2x2 − 10x − 28)/(x2 − 2x − 80)] ×[(5x2 − 36x − 32)/(6x2 − 12)]
• [(2( x2 − 5x − 14 ))/(x2 − 2x − 80)] ×[(5x2 − 36x − 32)/(6( x2 − 2 ))]
[(2( x + 2 )( x − 7 ))/(( x − 10 )( x + 8 ))] ×[(( 5x + 4 )( x − 8 ))/(6( x + 2 )( x − 2 ))]
Divide:
[(12x2y3z4)/(18xy4z)] ÷[(14x4yz2)/(81x3y3z3)]
• [(12x2y3z4)/(18xy4z)] ×[(81x3y3z3)/(14x4yz2)]
• [(6y2z2)/2y] ×[(9x2z2)/(7x2)]
• [(3yz2)/1] ×[(9z2)/7]
[(27yz4)/7]
Divide:
[(36a2b4c)/(27a4b3c2)] ÷[(48a4b2c3)/(54a3b5c2)]
• [(36a2b4c)/(27a4b3c2)] ×[(54a3b5c2)/(48a4b2c3)]
• [(3b2)/1a] ×[(2b2)/(4a2c2)]
[(6b4)/(4a3c2)]
Divide:
[(24j3k5i6)/(8j5k6i4)] ÷[(60j5k2i3)/(40j5k2i)]
• [(24j3k5i6)/(8j5k6i4)] ÷[(40j5k2i)/(60j5k2i3)]
• [(2i3)/(1k4i3)] ×[1/jk]
[2/(jk5)]
Divide:
[(4r − 28)/(10r − 40)] ÷[(7r − 49)/(3r + 15)]
• [(4r − 28)/(10r − 40)] ×[(3r + 15)/(7r − 49)]
• [(4( r − 7 ))/(10( r − 4 ))] ×[(3( r + 5 ))/(7( r − 7 ))]
• [4/(10( r − 4 ))] ×[(3( r + 5 ))/7]
[(12( r + 5 ))/(70( r − 4 ))]
Divide:
[(16t + 40)/(21t − 35)] ÷[(10t − 8)/(6t − 10)]
• [(16t + 40)/(21t − 35)] ×[(6t − 10)/(10t − 8)]
• [(8( 2t + 5 ))/(7( 3t − 5 ))] ×[(2( 3t − 5 ))/(2( 5t − 4 ))]
• [(8( 2t + 5 ))/7] ×[1/(5t − 4)]
[(8( 2t + 5 ))/(7( 5t − 4 ))]
Divide:
[(26x − 39)/(14x + 30)] ÷[(24x − 36)/(5x − 90)]
• [(26x − 39)/(14x + 30)] ×[(5x − 90)/(24x − 36)]
• [(13( 2x − 3 ))/(2( 7x + 15 ))] ×[(5( x − 18 ))/(12( 2x − 3 ))]
• [13/(2( 7x + 15 ))] ×[(5( x − 18 ))/12]
[(65( x − 18 ))/(24( 7x + 15 ))]
Divide:
[(6y − 20)/(16y + 18)] ÷[(36y − 120)/(8y − 36)]
• [(6y − 20)/(16y + 18)] ×[(8y − 36)/(36y − 120)]
• [(2( 3y − 10 ))/(2( 8y + 9 ))] ×[(4( 2y − 9 ))/(12( 3y − 10 ))]
• [1/(( 8y + 9 ))] ×[(2( 2y − 9 ))/6]
• [(2( 2y − 9 ))/(6( 8y + 9 ))]
[(2y − 9)/(3( 8y + 9 ))]
Divide:
[(2h − 10)/(4h + 8)] ÷[(2h2 − 8h − 10)/(2h2 + 7h + 6)]
• [(2( h − 5 ))/(4( h + 2 ))] ×[(2h2 + 7h + 6)/(2( h2 − 4h − 5 ))]
• [(2( h − 5 ))/(4( h + 2 ))] ×[(( 2h + 3 )( h + 2 ))/(2( h − 5 )( h + 1 ))]
• [1/4] ×[(2h + 3)/(h + 1)]
[(2h + 3)/(4( h + 1 ))]
Divide:
[(2x − x − 15)/(3x + 21)] ÷[(5x − 15)/(2x2 + 8x − 42)]
• [(2x − x − 15)/(3x + 21)] ×[(2x2 + 8x − 42)/(5x − 15)]
• [(( 2x + 5 )( x − 3 ))/(3( x + 7 ))] ×[(2( x2 + 4x − 21 ))/(5( x − 3 ))]
• [(( 2x + 5 )( x − 3 ))/(3( x + 7 ))] ×[(2( x + 7 )( x − 3 ))/(5( x − 3 ))]
• [(2x + 5)/3] ×[2/5]
[(2( 2x + 5 ))/15]
Divide:
[(14y + 42)/(14y + 35)] ÷[(2y2 + 4y − 6)/(2y2 + 13y + 20)]
• [(14y + 42)/(14y + 35)] ÷[(2y2 + 13y + 20)/(2y2 + 4y − 6)]
• [(14( y + 3 ))/(7( 2y + 5 ))] ×[(( 2y + 5 )( y + 4 ))/(2( y2 + 2y − 3 ))]
• [(14( y + 3 ))/(7( 2y + 5 ))] ×[(( 2y + 5 )( y + 4 ))/(2( y + 3 )( y − 1 ))]
• [7/7] ×[(y + 4)/(y − 1)]
[(y + 4)/(y − 1)]

*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.

### Multiply & Divide 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
• Objectives 0:09
• Multiply and Divide Rational Expressions 0:44
• Rational Numbers
• Dividing by Zero
• Canceling Extra Factors
• Negative Signs in Fractions
• Multiplying Fractions
• Dividing Fractions
• Example 1 8:04
• Example 2 14:01
• Example 3 16:23
• Example 4 18:56
• Example 5 22:43

### Transcription: Multiply & Divide Rational Expressions

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at how you can multiply and divide rational expressions.0003

There is a lot to know when it comes to rational expressions.0011

We will first start off by looking at what values would be undefined or restricted in some of these rational expressions.0013

We will also take quite a bit of time to figure out to how simplify them and put them into lowest terms.0021

There will be a few situations where you have to recognize that there are equivalent forms of these different rational expressions.0028

They may look different but they are the same.0035

Finally we will get into that multiplication and division process so you can see it all put together.0038

A lot of things that we will do these rational expressions ties into rational numbers in general.0046

In terms of the techniques and how we simplify them.0052

Remember that a rational number is any number that can be written as a fraction.0056

The number 2/3 is a good example of one of these rational numbers.0061

We are working with rational expressions you can consider those of the form p/q when p and q are both going to be polynomials.0066

Let me give you some examples of what these rational expressions look like.0076

2x + 3 ÷ 5x2 - 7.0080

Since both of those are polynomials and they are simply being divided that will be a good rational expression.0086

How about x3 - 27 ÷ 4x + 3? That will also be a good example of another rational expression.0093

What this rational expression since we have some variables in them, we are always concerned about dividing by 0.0107

We do not want use any values for x that would make the bottom 0.0113

Think of this expression up here x + 5 ÷ 2x - 4.0117

I can plug in many, many different values for my x as long as I do not get a 0 on the bottom.0123

In fact if I try and plug in a 2 into this that would give me 0 on the bottom.0130

2 × 2 - 4 and that is bit of a problem.0136

Since it gives me that 0 the bottom this expression is undefined at 2 or I might simply make a note somewhere and say x can not equal 2.0141

For many of the expressions that you will see on the future slides here,0151

we will assume that it can take on many different values, but not those restricted values.0156

When we are into the multiplying and dividing we will always be concerned with taking our answer down to its simplest form.0165

How is it that we go ahead and reduce a fraction to it simplest terms?0173

Remember how you do this process with some similar type of rational expressions.0178

We will try and cancel out a lot of the extra p’s that are present.0183

If I’m looking at 6x2 ÷ 2x2 I could look of this as 2 × 3 × x × x × x and in the bottom would be 2 × x × x.0188

As long as I have multiplication there I can simplify it by canceling out a lot of these extra factors and get something like 3x.0204

The good news is that type of simplifying is the same process we want to do for rational expressions.0215

We want to look for their common factors.0222

These are polynomials we will probably have to factor first, so that we have the actual factors in their p’s that are multiplied.0226

Watch how that works for this guy 4y + 2 ÷ 6y + 3.0233

Let us factor the top and bottom.0238

On the top there is a 2 in common and I can take it out from both parts.0241

That would be 2y + 1 and on the bottom it looks like there is a 3 in common, we will take that out, 2y + 1.0246

Notice we have this common piece 2y + 1 that is multiplied on the top and bottom.0259

We can cancel that out and we would just be left with 2/3.0266

Remember that we can only cancel out factors, pieces that are multiplied.0272

If you attempted to do some canceling at the very beginning, you can not do that, not yet because you are dealing with addition.0277

Be very careful as we try and simplify your expressions.0287

Some answers may look a little different from other answers, but actually they may just be equivalent.0294

One thing that can make equivalent expression is where you put the negative sign.0301

You could put it in the top, the bottom or out front of one of these rational expressions.0306

In all cases, they represent the same exact quantity.0313

You will say that they are all equivalent.0317

This also applies to your rational expressions, but sometimes it is not quite as obvious.0320

For example, maybe I'm looking at x - 7 ÷ 2.0328

Let us go ahead and put a negative sign on the top.0334

If I do that, that would be -x and may be distribute it through with that negative + 7 ÷ 2.0339

I could have also given that negative sign to the bottom x – 7 ÷ 2 and give that to the bottom and that would give me x - 7 ÷ -2.0350

These two expressions are equivalent.0367

They do equal the same thing, even though they look a little different.0369

Be careful if you are working on your homework, working with other people and you do not get exactly the same answer,0375

you might actually have equivalent expressions just inside a different form.0380

Now that we know about simplifying and some things to watch out for how do you get into the multiplication and division process.0389

I want to think back on how you do this with fractions.0396

Before multiplying fractions together and it is a nice process of multiplying across the top and multiplying across the bottom.0400

The same thing applies to your rational expressions.0409

We will be dealing with polynomials for sure, but will just multiply across the top and across the bottom.0412

In order for this to work, we must know how to factor our polynomials.0418

That way we can end up just multiplying their factors together.0423

When we are all done, we want to make sure that we have written it in lowest terms.0427

Try and cancel out any extra factors after you are done multiplying.0431

If you know how the multiplication process works, then you will also know a lot about the division process.0438

Think about how this works when you divide fractions.0445

From looking at something like 2/3 ÷ 5/7 and we have been taught to flip the second fraction and then multiply.0448

Which is of course, multiply across the top and multiply across the bottom.0456

There is some good news and this also applies to our rational expressions.0460

If you want to multiply them together, flip your second rational expression and then multiply the two.0465

Factoring will definitely help in this process that way we have to keep track of the individual factors, and where they go.0472

Always write your answer in lowest terms when you are all done.0479

That is quite a bit of information just on simplifying and multiplying and dividing.0485

We will look at some quick examples and see how this works out.0491

We want to take all of these rational expressions and put them into lowest terms.0497

We will be going through a simplification process.0501

Notice how in a lot of these we are dealing with addition and subtraction, do not cancel out yet until you get it completely factored.0505

I have (x2 - y2) ÷ (x2 + 2xy + y2).0514

These look like some very special formulas that we had earlier.0520

I have the difference of squares on top.0523

I have a perfect square trinomial on the bottom so I can definitely factor these.0525

I have x + y x – y on the top and on the bottom x + y, x + y.0533

Notice how we have a common factor the x + y.0549

We can go ahead and cancel that out.0554

This will leave us with an x - y ÷ x + y.0558

I can be assured that this is in the lowest terms because there are no other common factors to get rid of.0565

The next ones are very tricky.0571

Notice how the top and bottom almost look like the same thing.0574

It is tempting to try and cancel out.0578

Be careful, we cannot cancel them out unless they are exactly the same thing.0581

One thing to notice here is a -5 and here is 5.0586

Those are not the same thing, they are different in sign.0590

And same thing over here, this is w2 and this one is –w2.0593

Those are not the same in sign.0597

If you end up with a situation like this where they are almost the same, you are dealing with subtraction and the order is just reversed.0600

You can factor out a -1 from either the top or bottom.0607

If I factor out a -1 from the top then what is left over?0614

-1 × what will give me a w2, is a -w2 and let us see if I take out 5, that should do it.0620

-1 × -w = w, -1 × 5 = -5.0633

All of that is on top and I still have my 5 - w2 on the bottom.0641

We are getting a little bit closer and things are starting a matchup in sign a little bit better.0646

I’m just simply going to reverse the order of these and you will see that they are common factors.0652

-1 is still out front, (5 - w2) (5 + w2) and now I can go ahead and cancel these out.0657

The only thing left here is a -1.0677

That looks much nicer than what we started with.0680

In the next one I have 25q2 - 16/12 – 15q.0684

This one is going to take a little bit more work but I see I have one of those special cases on the top.0691

That is another difference of squares.0697

(5q + 4) (5q – 4)0700

Let us see if we can do anything with the bottom.0712

Does anything go into 12 and 15?0714

These both have a 3 in common, let us take that out.0718

We are looking pretty good and we can see that this is getting pretty somewhere to the previous example.0725

These look almost the same we are dealing with subtraction, but the order is just reversed.0732

We are going to take out a negative from the bottom so that they will be exactly the same.0736

(5q + 4) (5q – 4)0741

There you will take out the 3, let us take out -1 as well.0750

That will give us -4 + 5q.0753

It is better starting to match the top we just have to reverse the order.0758

Reading on the bottom 5q - 4 and now we can see we have a common factor to go ahead and get rid of.0773

The answer to this one would be 5q + 4 ÷ -3..0782

The last one involves 9 – t ÷ 9 + t.0791

In this one, another one that looks very close.0796

Unfortunately there is not a whole lot we can do to simplify it.0800

You might be wondering why cannot we just cancel out some 9 and call it good from there.0805

We can only cancel out common factors, things that are multiplied.0810

We cannot cancel out the 9 nor we can cancel out the t’s.0814

Unfortunately there is nothing to factor from the top or factor from the bottom.0818

This one is simplified just as it is.0822

Be on the watch out for cases like this and know when you can cancel out those extra terms.0834

In this next few we are going to go through the multiplication process and then try and bring it down to lowest terms.0844

We just have to multiply across the top and then multiply across the bottom.0850

That will make our lives a little bit easier.0854

Let us start off with this first one.0856

Multiplying across the top I will have 8x2 × 9 / 3xy2.0859

From here I can cancel a lot of my extra stuff.0873

I will cancel out extra 3 that it is the 9.0876

I can cancel out one of these x’s here and I can cancel out one of these y’s.0880

Let us see what is left over.0886

I still have 8x, 3, a single y on the bottom.0888

This is 24x ÷ y and that only multiply the two together, but I brought it down to lowest terms.0895

Onto the next one, multiplying the top together will give me 3t – (u × u) / (t × 2) × (t – u).0905

One obvious common term is that t – u, let us go ahead and get rid of that.0926

We would have left over 3u t × 2 or just make my brain feel better, 2 × t.0933

We have multiplied those together and reduce it to its lowest terms.0944

I want to point out something, back here it is tempting to go through the distribution process and put the 3 and t and the 3 and u,0949

but actually you do not want to do that just yet.0958

Go ahead and leave them into your factors because it will make it much easier to cancel them out.0961

If you do end up distributing them, you have to pull them back a part into their factors later on.0966

You are not saving yourself any work.0971

Leave the factors in there or if is not factored already go ahead and factor it so you can easier multiply and reduce.0974

Let us get into a much bigger one.0984

In this one we want to multiply together and then put into its lowest terms.0987

(x2 + 7x + 10 / 3x + 6) × (6x – 6 / x2 + 2x – 15)0992

This is a rather large one, but I'm not going to multiply together the tops first or the bottoms just yet.1005

I’m going to work on factoring just for little bit.1011

How would my first polynomial here factor?1017

My first terms would need to be an x that will give me x2 and my other terms would have to be 2 and 5.1023

That will be the only way that I can get that 10 and they would add to be the 7 in the middle.1031

That would factor that and we will also factor the bottom.1037

It looks like there is a common 3, let us pull that out.1041

We have just factored that first rational expression.1047

Onto the next one I see it has a common 6, x – 1 and on the bottom I see a trinomial.1051

Let us go ahead and break that down into two binomials.1061

X2 + 2x – 15.1065

What would that factor into?1068

Now that I have all the individual factors, I can multiply them together much easier.1075

I will simply write all of the factors on the top and multiply that out.1079

All the factors in the bottom, 3x + 2 x + 5 and x – 3.1092

In that step they are all multiplied together.1102

Let us cancel out a lot of those extra terms.1105

x + 5 were gone, x + 2 those were gone.1109

Let us cancel out one of the 3 and 6.1116

The only thing we have left over is 2 and x – 1 / x -3.1120

Since there are no more common factors I know that this one is finally in lowest terms.1129

This takes care of a lot of multiplication let us get into dividing these and also putting them into lowest terms.1138

With this one needs one need one extra step, we have to flip the second rational expression and then multiply and reduce.1144

Let us start off with this first one.1154

I have 9x2 ÷ 3x + 4.1156

I will flip the second one, 3x + 4 ÷ 6x3.1162

If I’m going to multiply the two together, I need to write all of their factors, the top ones together.1173

We will write all the factors of the bottom one together.1184

Now they are multiplied.1189

We can go ahead and cancel a lot of those extra terms.1192

We have a common 3x + 4 in the top and bottom, let us get rid of that, we do not need that.1195

We cancel out an extra 3 in the top and bottom.1201

On top we have x2 and in the bottom we have x3 so an x2 will cancel out leaving me x in the bottom.1207

That is quite a bit of canceling.1216

Let us see if we can figure out what is left over.1217

I still have a 3 on top, we still have a 2 and x on the bottom.1219

3 / 2x is the only thing left when we reduce it to lowest terms.1225

The next one looks like a do- see, let us go ahead and take this one nice and easy.1231

We will first write the first rational expression, just as it is.1237

We will go ahead and flip the second rational expression.1247

4r -12 and -r2 r + 31250

If I'm going to multiply this together we will go ahead and write all of the factors on the top.1263

4r2 – 1 on the bottom 2r + 1 -r2 and r + 31272

If we multiply things together let us see what we can cancel out.1290

One initial thing, I see an r + 3, let us get rid of that.1294

I got an r on top and r2 so let us get rid of some r.1299

I think there is even a little bit more we can cancel out as long as we recognize that we have a very special polynomial on top.1305

Notice how we have the difference of squares on top the 4r2 -1.1318

We can write that as 2r + 1 and 2r - 1.1325

We can see that we do have an extra factor hiding in there that we can get rid of.1333

We can get rid of that 2r + 1.1338

What is left over, I have 4 × 2r -1 on the top / -r in the bottom.1346

This one is completely factored.1354

I got one more example and this is another large one.1365

We will have to just walk through it very carefully.1369

I have xw - x2 / x2 – 1 then we are dividing that by x - w / x2 + 2x +1.1371

Let us see what we can do.1381

Let us go ahead and rewrite it and have the second one flipped over.1383

Multiply it by x2 + 2x + 1x – w.1395

We will go ahead and start putting things together.1407

Let us multiply (xw - x2) (x2 + 2x + 1) ÷ (x2 – 1) (x – w)1410

This one looks like it is loaded with lots of things that we can go ahead and factor.1428

Let us go ahead and do that and we are going to do that bit by bit.1432

I’m looking at this very first factor here and notice how they both have terms in there that have x.1436

We can factor out an x from each of those.1443

That will leave us with w – x.1449

In this one over here we can factor it into an x + 1 and x + 1.1454

It is one of our perfect squares.1460

(x + 1)(x +1)1465

On to the bottom, this one right here is the difference of squares x + 1 x -1.1473

Unfortunately this guy we do not have anything special about it, I will just write it.1486

That will allow me to at least cancel out a few things.1493

I have an extra x + 1 that I will go ahead and get rid of.1496

It looks like I can almost get rid of something else.1500

Notice how we have this w – x and x – w.1503

It is almost the same thing, and it is involving subtraction.1507

I need to factor out a -1 from one of these.1511

Let us do it to the top -w + x.1515

I still have an x + 1 on top / x – 1 and x – w.1522

-x will rearrange the order of these guys and switch them around.1537

That will give us (x – w) (x + 1) / (x – 1) (x – w).1544

We can see sure enough, there is another piece that we can go ahead and get rid of.1556

We only have one thing left, -x × x +1 / x – 1.1563

Since we have no other common factors, this is finally in its lowest terms.1575

When it comes to working with these rational expressions, remember how all of these processes work, which is normal fractions.1580

Whether that means multiplying fractions by going across the top and bottom1587

or dividing fractions by flipping the second one.1591

Then cancel out your extra factors so you can be assured that is brought down to lowest terms.1593

Thank you for watching www.educator.com.1599

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