# The Greatest Common Factor And Factoring By Grouping Worksheet With Answer Key - Section 6.1

Section 6.1 The Greatest Common Factor and Factoring by Grouping
Greatest Common Factor (GCF):
 The GCF is an expression of the highest degree that divides each term of the polynomial.
 The variable part of the greatest common factor always contains the smallest power of a
variable that appears in all terms of the polynomial.
Finding GCF
 Step 1: Find the prime factorization of all integers and integer coefficients
 Step 2: List all the factors that are common to all terms, including variables
 Step 3: Choose the smallest power for each factor that is common to all terms
 Step 4: Multiply these powers to find the GCF
NOTE: If there is no common prime factor or variable, then the GCF is 1
4
3
5
2
(Example) Find the GCF for the following set of algebraic terms: {30x
y, 45x
y, 75x
y
}
4
4
30x
y =
2·3·5·x·x·x·x·y
=
2·3·5·x
y
3
2
3
45x
y =
3·3·5·x·x·x·y
=
3
·5·
x
y
30
45
75
5
2
2
5
2
75x
y
= 3·5·5·x·x·x·x·x·y·y =
3·5
·x
y
3 ∙ 5 ∙
15
Therefore, GCF is
2
15
3
15
3
25
3
5
3
5
5
5
¿
?
Negative GCF
 In my opinion, we do not need to factor out a negative GCF
 Factoring out a negative GCF is exactly same as factoring out GCF, but last step.
 Step 1: Find the prime factorization of all integers and integer coefficients
 Step 2: List all the factors that are common to all terms, including variables
 Step 3: Choose the smallest power of each factor that is common to all terms
 Step 4: Multiply these powers to find the GCF
Factoring by Grouping will be discussed with examples
Exercises
(Solution 1)
Step 1: Prime Factorization (ignore signs)
18
∙ 3 ∙ 3
∙ 3
8
∙ 2 ∙ 2
10
∙ 5
∙ 5
Step 2: List all factors that are common to all terms
Common factors to all are 2 and y
Step 3: Choose the smallest power
The smallest power for 2 is 1
The smallest power for y is 1
So, the GCF is 2 ∙
2
Cheon-Sig Lee
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