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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

Step 5: Put negative sign in front of 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

Page 1

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Parent category: Education