# 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, x^{3} is the greatest common factor of x^{3}
and x^{5} . (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: 15x^{3} + 9x^{2}

** Solution :** The GCF is 3x^{2} (3 is the common factor between 15
and 9 and x^{2} is the

common factor between x^{3} and x^{2} . We factor out 3x^{2
} from each term to get

3x^{2} (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**

Factor the Greatest Common Factor (GCF)

Factor by grouping

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