Greatest Greatest Common FactorCommon Factor

Greatest Common FactorGreatest Common Factor

The greatest common factor The greatest common factor (GCF) is the product of the (GCF) is the product of the prime factors both numbers prime factors both numbers have in common.have in common.

OrOrIt is the largest number that is It is the largest number that is a factor of all original a factor of all original numbers.numbers.

Find the Greatest Common Find the Greatest Common FactorFactor

Example: 18xy , 36y2

18xy = 2 · 3 · 3 · x · y

36y2 = 2 · 2 · 3 · 3 · y · y GCF = =

18y2 · 3 · 3· y

Tips for finding the GCFTips for finding the GCF

Find the prime factorization of Find the prime factorization of each item.each item.

Circle what is common.Circle what is common. Multiply together what is Multiply together what is

common to get the GCF.common to get the GCF.

Now you try!Now you try!

Example 1:Example 1: 12a12a22b , 90ab , 90a22bb22cc

Find the greatest common factor of the following:

Example 2:Example 2: 15r15r22 , 35s , 35s22 , 70rs , 70rs

GCF = 6a2b

GCF = 5

Last ExampleLast Example

What is the greatest What is the greatest common factor of 15ab common factor of 15ab and 16c?and 16c?

Factoring Factoring Using the GCFUsing the GCF

FactoringFactoring- ““Undoing” distributionUndoing” distribution- Finding factors that, Finding factors that, when multiplied, form when multiplied, form the original polynomialthe original polynomial

Example:Example:

Factor:Factor: 12a12a22 + 16a + 16a

= 2·2·3·a·a + = 2·2·3·a·a + 2·2·2·2·a2·2·2·2·a= = 22

· · 22· · aa(3·a + 2·2)

= 4a (3a + 4)

You can check by distributing.

1. Factor each term.

2. Factor out the GCF.

3. Multiply.

Example:Example:

Factor: Factor: 18cd18cd22 + 12c + 12c22d + 9cdd + 9cd

= 2·3·3·c·d·d + 2·2·3·c·c·d + = 2·3·3·c·d·d + 2·2·3·c·c·d + 3·3·c·d3·3·c·d= = 33 · · cc · · dd (2·3·d + 2·2·c + 3)

= 3cd (6d + 4c + 3)

Now you try!Now you try!

Example 1:Example 1:

15x + 25x15x + 25x22

Example 2:Example 2:

12xy + 24xy12xy + 24xy22 – 30x – 30x22yy44

= 6xy(2 + 4y – 5xy3)

= 5x(3 + 5x)

One last example:One last example:

Factor:Factor: 4x + 12x4x + 12x22 + 16x + 16x33

= 2·2·x· + 2·2·3·x·x + = 2·2·x· + 2·2·3·x·x + 2·2·2·2·x·x·x2·2·2·2·x·x·x= = 22 · · 22· · xx(1 + 3·x + 2·2·x·x)

= 4x (1 + 3x + 4x2)