  Eric Smith

Greatest Common Factor & Factor by Grouping

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: Sha TaoWed Apr 1, 2020 8:08 AMPost by Sha Tao on April 1, 2020In factor by grouping, 6(p+q)- r(p+q) is  6p+6q-rp-rq. You forgot the minus sign before rq. 1 answer Last reply by: Professor Eric SmithWed May 15, 2019 8:32 PMPost by Austin An on May 15, 2019In Number 5 did you forget to carry the exponent when dividing because I thought 6w^2 / 3w = 2w^2 3 answersLast reply by: John StedgeThu Jun 22, 2017 10:54 AMPost by Destiny Coleman on September 23, 2014I did the work on Example 6 differently so I'm wondering if my answer works.  First I rearranged differently.  Instead of changing to: 9xy+12x-3y-4 I used 9xy-3y+12x-4This changed to -3y(-3x+1)-4(-3x+1)The answer that I recieved was (-3x+1)(-3y-4)

### Greatest Common Factor & Factor by Grouping

• Factoring means to rewrite an expression as a product. Think of it as the opposite of multiply things together.
• The greatest common factor is the greatest quantity that would even divide into all of the terms. This could be made up of numbers and variables.
• If a polynomial has only one as its greatest common factor it is said to be prime.
• To factor by grouping
• Collect your terms into two groups
• Factor out the greatest common factor of each group
• Factor the entire polynomial, noting that they now have a common factor
• Rearrange and try again if they do not have a common factor

### Greatest Common Factor & Factor by Grouping

Write in prime factored form: 400
• 400 = 40 ×10
• 8 ×5 ×2 ×5
• 2 ×2 ×2 ×5 ×2 ×5
2 ×2 ×2 ×2 ×5 ×5
Write in prime factored form: 224
• 224 = 4 ×56
• 2 ×2 ×8 ×7
2 ×2 ×2 ×2 ×2 ×7
Write in prime factored form: 144
• 144 = 12 ×12
• 3 ×4 ×3 ×4
• 3 ×2 ×2 ×3 ×2 ×2
2 ×2 ×2 ×2 ×3 ×3
Write in factored form:
196a4b3c
• 4 ×49 ×a ×a ×a ×a ×b ×b ×b ×c
2 ×2 ×7 ×7 ×a ×a ×a ×a ×b ×b ×b ×c
Write in factored form:
175g2hi5
• 25 ×7 ×g ×g ×h ×i ×i ×i ×i ×i
5 ×5 ×7 ×g ×g ×h ×i ×i ×i ×i ×i
Write in factored form:
48u2v3w
• 4 ×12 ×u ×u ×v ×v ×v ×w
• 2 ×2 ×3 ×4 ×u ×u ×v ×v ×v ×w
2 ×2 ×2 ×2 ×3 ×u ×u ×v ×v ×v ×w
Find the GCF of 24, 60, and 72
• 24 = 2 ×2 ×2 ×3
• 60 = 5 ×12 = 2 ×2 ×3 ×5
• 72 = 6 ×12 = 2 ×3 ×2 ×2 ×3 = 2 ×2 ×2 ×3 ×3
• 24 = 2 ×60 = 5 ×12 = ×572 = 6 ×12 = 2 ×3 ×2 ×2 ×3 = 2 ××3
GCF = 2 ×2 ×3 = 12
Find the GCF of 36, 50, and 92
• 36 = 6 ×6 = 2 ×2 ×3 ×3
• 50 = 2 ×25 = 2 ×5 ×5
• 92 = 2 ×46 = 2 ×2 ×23
• 36 = 6 ×6 = ×2 ×3 ×350 = 2 ×25 = ×5 ×592 = 2 ×46 = ×2 ×23
GCF = 2
Find the GCF of:
12x2y2, 18xy3, 72x3y
• 12x2y2 = 2 ×2 ×3 ×x ×x ×y ×y
• 18xy3 = 2 ×9 = 2 ×3 ×3 ×x ×y ×y ×y
• 72x3y = 6 ×12 = 2 ×3 ×2 ×2 ×3 = 2 ×2 ×2 ×3 ×3 ×x ×x ×x ×y
• 12x2y2 = ×2 ×××x ××y18xy3 = 2 ×9 = ××3 ×××y ×y72x3y = 6 ×12 = 2 ×3 ×2 ×2 ×3 = ×2 ×2 ××3 ××x ×x ×
• 2 ×3 ×x ×y
GCF = 6xy
Find the GCF of:
16g5h3i, 36g2h3i4, 48g3h2i4
• 16g5h3i = 2 ×8 = 2 ×2 ×2 ×2 ×g ×g ×g ×g ×g ×h ×h ×h ×i
• 36g2h3i4 = 3 ×12 = 3 ×3 ×2 ×2 = g ×g ×h ×h ×h ×i ×i ×i ×i
• 48g3h2i4 = 4 ×12 = 2 ×2 ×2 ×2 ×3 ×g ×g ×g ×h ×h ×i ×i ×i ×i
• 16g5h3i = 2 ×8 = ××2 ×2 ××g ×g ×g ××h ×36g2h3i4 = 3 ×12 = 3 ×3 ×× = ××h ××i ×i ×i48g3h2i4 = 4 ×12 = ××2 ×2 ×3 ××g ×××i ×i ×i
• 2 ×2 ×g ×g ×h ×h ×i
4g2h2i
Factor:
4x2y3z4 − 12x2y2z + 18xy3
• GCF = 2xy2
2xy2( 2xyz4 − 6xyz + 9y )
Factor:
13a3b4c + 26a2bc3 − 39ab3c5
• GCF: 13abc
13abc( a2b3 + 2ac2 − 3b2c4 )
Factor:
8r7s5t4 − 16r6s3t2 − 48r8s6t4
• GCF: 8r6s3t2
8r6s3t2( rs2t2 − 2 − 6r2s3t2 )
Factor:
( 5x2 − 10xy + 4xy − 8y2 )
• ( 5x2 − 10xy )
GCF: 5x
• ( 4xy − 8y2 )
GCF: 4y
• 5x( x − 2y ) + 4y( x − 2y )
( 5x + 4y )( x − 2y )
Factor:
( 6x2 + 9xy − 14xy − 21y2 )
• ( 6x2 + 9xy )
GCF: 3x
• ( − 14xy − 21y2 )
GCF: ( − 7y )
• 3x( 2x + 3y ) + ( − 7y )( 2x + 3y )
( 3x − 7y )( 2x + 3y )
Factor:
( 13s2 − 39st − 25st + 75t2 )
• ( 13s2 − 39st )
GCF: 13s( s − 3t )
• ( − 25st + 75t2 )
GCF: 25t( − s + 3t ) = 25t( s − 3t )
• 13s( s − 3t ) − 25t( s − 3t )
( 13s − 25t )( s − 3t )
Factor:
16x3 − 4x2 − 3 + 12x
• ( 16x3 − 4x2 ) + ( − 3 + 12x )
• 4x2( 4x − 1 ) + 3( − 1 + 4x )
• 4x2( 4x − 1 ) + 3( 4x − 1 )
( 4x2 + 3 )( 4x − 1 )
Factor:
34y4 − 17y2 − 24y + 48y3
• ( 34y4 − 17y2 ) + ( − 24y + 48y3 )
• 17y2( 2y2 − 1 ) + 24y( − 1 + 2y2 )
• 17y2( 2y2 − 1 ) + 24y( 2y2 − 1 )
( 17y2 + 24y )( 2y2 − 1 )
Solve:
( 6m + 4 )( 5m − 10 ) = 0
• 6m + 4 = 0
• 6m = 4
• m = [4/6] = [2/3]
• 5m − 10 = 0
• 5m = 10
• m = 2
{ [2/3],2 }
Solve:
( 2a − 14 )( 6a + 36 ) = 0
• 2a − 14 = 0
• a = 7
• 6a + 36 = 0
• 6a = − 36
• a = − 6
{ − 6,7 }

*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.

### Greatest Common Factor & Factor by Grouping

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
• Greatest Common Factor 0:31
• Factoring
• Greatest Common Factor (GCF)
• GCF for Polynomials
• Factoring Polynomials
• Prime
• Example 1 9:14
• Factor by Grouping 14:30
• Steps to Factor by Grouping
• Example 2 17:43
• Example 3 19:20
• Example 4 20:41
• Example 5 22:29
• Example 6 26:11

### Transcription: Greatest Common Factor & Factor by Grouping

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at the greatest common factor and the technique of factor by grouping.0002

First we will learn how to recognize a greatest common factor for a list of different terms0012

and how we can actually factor out that greatest common factor when you have something like a polynomial.0017

This will lead directly into the technique of factor by grouping, a good way of breaking down a large polynomial.0023

You are going to hear me use that word factor quite a bit in this entire section.0032

You are probably a little curious about what exactly factoring means.0036

Technically it means to write a quantity as a product.0043

We will be breaking down into pieces that are multiplied.0047

In a more practical sense, the way I like to think of factoring is that you are ripping things apart.0051

It is almost like doing the opposite of multiplication.0056

For example if I have 2 × 3 I could put them together and get 6.0059

When I have the number 6 then I can factor it down into those individual pieces again 2 × 3.0066

It is like multiplication but we are going in the reverse direction.0072

Now some expressions and numbers could have many different factors.0077

Let us take the number 12.0082

If you break that one down, you could look at it as 2× 6 and you could continue in breaking down the 6 and 2 × 3.0085

You have factors like 2 would go in there, 4 would go into 12, 3 would be a factor, 6 would be factor.0095

All those things could go into 12.0103

The greatest common factor will be the largest number that will divide into all the numbers present.0110

I will show you this to you twice just to get a better feel for it.0116

Suppose I'm looking at 50 and 75, the greatest common factor would be the largest number that could divide into both of those evenly.0120

5 could divide it into both of those evenly and that would be a pretty good choice, but is not the largest thing that could go in there.0128

The largest thing that could go in would be the number 25.0136

You will see that if I do take them both and divide them by 25, true enough they both go in there evenly.0140

25 is my greatest common factor.0150

We may only do this for pairs but you could do it with an entire list of numbers.0155

In this one I have 12, 18, 26, 32, and again we are looking for the largest thing that divides into all of that.0162

Sometimes a good way to find the largest thing that goes into all of them is just find something that works,0169

maybe something like 2 and end up reducing them bit by bit.0177

If I divide everything in here by 2 I would get 6, 9, 13 and 16.0183

Nothing else will go into all of those since 13 is a prime number.0191

My greatest common factor is 2 in this case.0197

It is not a very big number but it happens to be the largest in there that will go into all for these numbers evenly.0201

In algebra we are not interested in just single numbers we also want to see this process work out when we have lots of variables.0209

Let us see how this works out.0216

Maybe I’m looking at the numbers over the terms -16r9, -10r15, 8r12.0218

To find the greatest common factor, I’m first going to look at those numbers and see if there is a large number that could go into all of them.0226

I could divide all of them by 2, that would work but actually I think 2 is the largest one.0234

I will say that 2 will divide it into all of these evenly.0242

Divided by 2, divided by 2, divided by 2 and I would get -8, -5, 4.0248

We can also do this thinking about our variables.0258

What was the largest variable raise to the power that we can divide into all of them?0261

Think of using your quotient rule to help out.0266

With this one r9 is the greatest thing that I can divide all of them by.0272

I would not have to deal with negative exponents or anything like that.0279

Let us write that over here, r9.0284

r9 ÷ r9 would be a single r0 or 1.0288

r15 ÷ r9 = r6 and r12 ÷ r9 = r3.0293

Our greatest common factor would be 2r9.0303

Be careful when you are dealing with those variables and exponents.0307

Sometimes you will hear greatest common factor in your mind and you think you should go after the greatest exponent.0310

But as we can see in this example, it is not the case.0315

We actually took the smallest exponent because it was simply the greatest thing that could go into all of them.0319

Put that in mind and let us try this again.0327

In this one I have nothing but variables.0330

h4 k6 h3 k6 and h9 k2.0332

What is the largest power of h that could go into all of them?0340

It is going to be h3.0345

I picked up on that because I can see that looking in the middle one it is the smallest power.0349

Looking at all the k’s, the largest exponent of k that could go into all them would be k2.0354

My greatest common factor will be h3 k2.0364

Let us see what this will end up reducing down to if I take that greatest common factor.0373

h1 k4 h3 and h3 will cancel each other out.0378

k4 I have h6 and I have no k that will end up cancelling each other out.0388

Being able to recognize the greatest common factor will help us in the next process.0396

Make sure it is pretty solid.0400

If you can recognize the greatest common factor you can often factor it out of a polynomial.0405

What I mean by factoring out is we are going to place it out front of a pair of parentheses0413

and put the reduced terms inside of that parentheses .0418

3m + 12 this is like looking at 3m and looking at 12, what is the greatest common factor of those two terms?0423

What could potentially go into both of them evenly and what is the largest thing that will do that?0431

The number 3 will go into 3m and the number 3 will also go into 12.0438

3 will be my greatest common factor.0446

I’m going to write that out front on the outside of the parenthesis.0449

What I write on the inside, what happens when I take 3m ÷ 3?0453

I will get just single m and when I take 12 ÷ 3 I will get 4.0458

Here I have taken a polynomial and essentially factored it into a 3 and into another polynomial m + 4.0467

The neat part about this is since factoring is like the opposite of multiplication you can check this by running through the distribution process.0475

If you take 3 and you put it back in here do you get the original answer?0487

You will see that in fact you do get 3m + 12.0492

That is how you know this has been factored correctly.0496

In some instances, just like with numbers when you are looking for the greatest common factor0501

it turns out you would not be able to find greatest common factor to pull from all of the terms or the largest thing will actually be 1 or itself.0506

In cases like that, we say that the polynomial is prime.0513

In other words it does not break down into any other pieces.0517

Let me give you a quick example of a prime polynomial 5x + 7.0520

There is not a number that goes into 5 and into 7 so I would consider that one prime.0526

Individually those numbers are divisible, but there is not a number that goes in the 8 and 9 it would be a prime polynomial.0542

Let us work on factoring out the greatest common factor from various different polynomials,0555

just so we get lots of good practice with it.0560

Starting with the first one I have 6x4 + 12x2.0562

Let us hunt down those numbers first.0567

The largest number that would go into both 6 and 12 would be 6.0570

Let us look at those x.0578

What is the largest power of x that would divide into both of them?0581

That have to be x2.0586

We will write in the leftovers inside our parentheses 6x4 ÷ 6x2=x2 and 12x2 ÷ 6x2 = 2.0592

I factored out the greatest common factor for that polynomial.0611

Let us give this another shot with something has a bunch of terms to it.0615

30x6 25x5 10x4.0619

Looking at the numbers 30, 25, and 10, what is the largest thing that could go into all of them, 5 will do it.0625

Looking at our variables the largest variable that could go into all them would be x4.0634

It is time to write down what I left over 30 ÷ 5 = 6, x6 ÷ x4 = x2.0645

25x5 ÷ 5 = 5, x5 ÷ x4 =x and 10x4 ÷5x4 = 2.0659

That one has been factored out.0675

Continuing on, let us get into some of the trickier ones.0678

This one has 3 terms and has multiple variables in there.0681

It has e and t.0685

Let us look at our numbers.0688

What could go into 8, 12 and 16?0689

2 would definitely go into all of them.0696

4 is larger and it would also go into all of them and I think that is largest thing, so we will go with 4.0698

Let us take care of these variables, one at a time.0707

Looking at the e’s, what is the largest power of e that could go into all of them?0710

This will be e4.0715

Onto the t’s, the largest exponent of t that would go into all of them is t2.0721

Now that we have properly identified our greatest common factor, let us write down what is left over.0732

8 ÷ 4 =2, e5 ÷ e4 =e and t2 ÷ t2 =1.0737

16 ÷ 4 = 4, e6 ÷ e4 = e2 and t3 ÷ t2 =t.0752

Onto the last one, -12 ÷ 4 = -3, e4 ÷ e4 = 1 and t7 ÷ t2 = t5.0766

We have our factor polynomial.0790

One last one, I threw this one in so we can deal some fractions.0794

We will first think of what fraction could divide into ¼ and into ¾ ?0800

¼ is the only thing that will do it.0808

What could go into y9 and y2?0812

Look for the small exponent that usually helps out.0817

That will be our greatest common factor, let us write what is left over.0823

¼ ÷ ¼ = 1 and y9 ÷ y2 = y7.0828

¾ ÷ ¼ = 3 and y2 ÷ y2 = 1.0839

At all of these instances you identify your greatest common factor first and simply write your leftovers inside the parentheses.0849

One quick thing that can help out, you can double check your work by simply multiplying your greatest common factor0856

back into those parentheses using your distribution property.0861

If it turns out to be the original problem, you will know that you have done it correctly.0865

In some terms, there is a much larger piece in common that you can go ahead and pull out.0873

Even though it is larger, feel free to still factor it as normal.0880

That is the hard part.0884

To demonstrate this I have 6 × (p + q) –r (p + q).0887

It has that p + q piece again and both of its parts.0894

When I'm looking at the greatest common factor of this first piece and the second piece, it is that p + q.0899

I’m going to take out the p + q as an entire piece and that is my greatest common factor.0910

Inside the set of parentheses I will write what is left over.0919

From the first part there is a 6 that was multiplied by p + q so I will put that in there and there was a -r on the second piece.0924

This looks a little strange, I mean it seems weird that we can take out such a large piece, but unusually that it does work.0933

To convince you I will go ahead and multiply these things back together using foil.0941

What I have here is my first terms would be 6p, outside terms would be –pr.0947

Inside terms 6q and my last terms –qr.0955

Compare this after I use my distribution property on the original.0962

(6p + 6q) – (rp + rq).0970

What you see is that it does match up with the original.0984

There is my 6p, I have my – pr and – rp, there are just in a slightly different order.0988

I have 6q and 6q -qr and I forgot to distribute my negative sign.0995

This should be a -rq and so it is the same term.1007

What you find is that you can pull out that large chunk of p + q and that leads to what is known as factor by grouping.1013

The idea behind factor by grouping is you try and take out the greatest common factor from a few terms at a time.1024

Rather than looking at all of them, just take them in parts.1031

This grades usually a very large piece and you can collect it into two groups.1034

Then you factor within those groups, you take out one of those large pieces.1040

This will allow you to factor the entire polynomial.1045

Sometimes factor by grouping does not seem to work or usually you end up struggling with a little bit.1048

If that is the case, try rearranging the terms and try to factor by grouping one more time.1053

It is a neat process and let us give it a try, when you factor by grouping on p × q + 5q + 2p +10.1060

Watch how this works.1071

Rather than looking at all the terms at once, I’m going to take them two at a time.1073

A part of my motivation for doing that is if I were to look at all the terms all at once, they do not have anything in common.1078

Let us take these first two.1086

What would be the greatest common factor for p × q + 5q.1088

q is in both of the terms so that is my greatest common factor for both of those.1094

What is left over? There is still a p in there and there is still 5.1101

Those are done.1109

Let us look at the next two terms.1111

What is the greatest common factor of just those two.1115

I can see a 2 goes into both of them and then let us see, the only thing left over would be a p + 5.1119

I will factor them into those little individual groups and now notice how I only have two things and they both have a p + 5 in common.1131

I'm going to take out the p + 5 as my greatest common factor of those two terms.1142

What will be left behind will be the q + 2.1151

This will represent my factor polynomials.1156

Let us try this again and see how it could work.1162

I have 2xy + 3y + 2x + 3.1164

Looking at the first two.1170

These have a y in common, let us go ahead and take that out.1174

2x will still be left when I divide 2xy by y and I still have a 3 over here when I divide 3y by y.1187

Looking at the next two terms, it looks like these ones do not have anything in common.1198

I might consider that the only thing that you do have in common is just a 1.1205

I could divide them both by 1.1209

I still have a 2x and I still have a 3.1213

Notice how we have that common chunk in there 2x + 3.1218

We will take that out 2x + 3 and then we will write what is left over just the y +1.1223

We can consider this one factored.1237

In this next example, be on the watch out for some negative signs which could show up as we factor out that greatest common factor.1242

I’m going to take this two at a time.1251

Looking at the first two, they do not have a number in common, but they both looks like can be divided by x2.1255

Let us take that out.1262

After taking our x2 we would have left over a single x and a 3.1268

Let us look at the next two terms.1280

I can see that if I would go into -5 and 5 would divide into a -15 and they both have the negative sign.1284

That is important because I’m thinking ahead and try to think how I also have this common x + 3 piece.1292

Especially if I’m trying to factor by grouping.1300

The only way that is going to work out is if I take out a greatest common factor that is a -5.1303

What would that does to our left over pieces now?1310

-5x ÷ -5 = x and -15 ÷ -5 = 3.1314

I get those left over pieces like I need to, that same x +3 on the other side.1324

Now I have this I can take out my common piece of x + 3 and I can go ahead and write the leftovers x2 – 5.1332

Be on the watch out for certain situations where you need to rearrange things first.1352

In this one I have 6w2 - 20x + 15w - 8wx.1357

And it is tempting to just go ahead and jump in there and try and factor.1364

Watch what happens if you do so.1367

First we take the first two and we would look for something common.1370

The only thing I can see that would be common is a 2 goes into 6 and into 20 and they do not have the same variables.1375

That is all I can do.1385

I have 3w2 and -10x.1386

Looking at the other terms the only thing that I see common over there is they have a w in common.1393

What would be left over? I still have a 15 and have 8x.1406

That is definitely a problem because these pieces are not the same.1413

I need to do some rearrangement and retry this factor by grouping one more time.1419

Let us go ahead and rewrite this.1433

I will rewrite it as 6w2 + 15w - 8wx and we will put that -20x on the very end.1435

Here is a little bit of my motivation for putting in that order.1452

When looking at the w's, notice how this one is a w2 and this one is a single w.1456

I have the one is one more in power after the left.1460

And looking at the same w's, here is my w1, this is like w0.1464

It is like I have lined things up according to the w power.1470

Watch how I factor by grouping works out much better.1475

Taking the first two they both have a w in common and I can pull that out.1481

It also looks like I can pullout 3.1485

3w will be like our greatest common factor.1488

What would be left over on the inside?1494

6w2 ÷ 3w = 2w and 15w ÷ 3w = 5.1499

I have -8wx -20x, both of those are negative, I think I will go ahead and pull out a -4, that will go into both.1510

They both contain x so we will also take out an x.1523

-8wx ÷ -4x.1532

There are still 2w left in there.1536

-20x ÷ -4 = 5.1540

I can see there is that common piece that I wanted the first time.1546

We can go ahead and factor that out front.1552

2w + 5 and 3w - 4x and now this is completely factored.1556

Let us do one last one, and this is the one where we might have to do a little bit of rearranging just make sure it works out.1575

9xy – 4 + 12x – 3y.1581

What to do here?1587

Let us go ahead and put the things that have x together.1590

9xy + 12x - 3y - 4 and I have just rearranged it.1606

We will look at these first two terms here and take out their greatest common factor.1612

You can see that 3 goes into both of them and they both contain x.1618

It will take both of those out.1624

9x ÷ 3x, 9 ÷ 3 = 3, x ÷ x =1 and we still have a y sitting in there.1628

12 ÷ 3 = 4, x ÷ x = 1.1639

Those two would be gone.1643

That looks pretty nice and it is actually starting to match what I have over here.1646

Notice that the only difference is a negative.1651

I'm going to take out a -1 from both of the terms that not should be able to flip my signs and make it just fine.1655

-3y ÷ -1 = 3y and -4 ÷ -1 = 4.1663

I have that nice common piece that I need.1672

We will go ahead and factor that out .1677

3y + 4 and 3x - 1.1681

Now this is completely factored.1690

When using the technique of factor by grouping take it two at a time and factor those first.1692

Look for your common piece and factor that out and that should factor your entire polynomial completely.1698

Thank you for watching www.educator.com.1705

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