 # GCF and Factoring by Grouping

The greatest common factor , or GCF, is the largest factor each term has in common. The
GCF can include numbers and variables . In terms of numbers, it is the largest factor each
number has in common . For example, 4 is the greatest common factor of the two
numbers 4 and 20. (Notice that 2 is also a common factor ; however, it is not the greatest
common factor.) In terms of variables , the GCF is the largest exponent each variable has
in common
. For example, x3 is the greatest common factor of x3 and x5 . (Again, x is a
common factor; however, it is not the greatest common factor).

We can factor a polynomial using the GCF (this means we are going to do a reverse
distributive property ). Remember distributive property means multiplication, so the
reverse is division . Factoring is a form of division.

Example 1: Factor out the GCF: 15x3 + 9x2

Solution : The GCF is 3x2 (3 is the common factor between 15 and 9 and  x2 is the
common factor between  x3 and x2 . We factor out 3x2  from each term to get
3x2 (5x + 3 ). You can check your answer by performing distributive property (you
should get the original problem).

Example 2: Factor out the GCF: Solution: The GCF is 6xy . We will factor 6xy from each term to get Factoring by grouping is used when there is four terms in the polynomial. We will group
the first two terms and factor out the GCF then group the next two terms and factor out
the GCF. We will have gone from four terms to two terms and factor out what is
common. Again, we will be doing reverse distributive property.

Example 3: Factor Solution: There are four terms; therefore, we will separate the first two terms from the
next two terms and find the GCF of each pair . Example 4: Factor Solution: Again, there are four terms; therefore, we will factor by grouping. Any time
you factor by grouping, it is not a coincidence the two terms have the same factor in
parentheses. You always want the same binomial in parentheses in the second step . Practice Problems

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