Factoring By Grouping
After studying this lesson, you will be able to:
Steps of Factoring:
1. Factor out the GCF
2. Look at the number of terms:
 2 Terms: Look for the Difference of 2 Squares
 3 Terms: Factor the Trinomial
 4 Terms: Factor by Grouping
3. Factor Completely
4. Check by Multiplying
This lesson will concentrate on the second step of factoring:
Factoring by Grouping.
**When there are 4 terms, we use factoring by grouping**
Example 1
Factor 3xy  21y + 5x  35
We have 4 terms, so we use factoring by grouping.
The first thing we do is the group the first 2 terms together
and group the last 2 terms together. We do this by inserting
parentheses.
( 3xy  21y ) + ( 5x  35 )
Now we work with the groups. We factor out the GCF in the
first group and the GCF in the second group: 3y is the first GCF
and +5 is the second GCF:
3y ( x  7) +5 ( x 7 )
Now, we look at this as 2 terms... 3y ( x  7 ) is the first
term and 5 ( x  7 ) is the second term.
We factor out what these 2 terms have in common  in this case
they have x  7 in common
So, we factor out x  7 and we will have ( x  7 ) ( 3y + 5 )
this is the answer
The 3y + 5 is what is left after we factored out x  7
We can check the answer by multiplying ( x  7 ) ( 3y + 5 )
use the FOIL method
3xy +5x 21y 35 this answer matches the original problem
(order doesn't matter when adding)
Example 2
Factor 8m^{ 2} n  5m  24mn + 15
We have 4 terms, so we use factoring by grouping.
The first thing we do is the group the first 2 terms together
and group the last 2 terms together. We do this by inserting
parentheses.
( 8m^{ 2} n  5 )( m  24mn + 15 )
Now we work with the groups. We factor out the GCF in the
first group and the GCF in the second group: m is the first GCF
and 3 is the second GCF:
m ( 8mn  5 ) (3 ( 8mn  5 ))
Now, we look at this as 2 terms... m ( 8mn  5 ) is the first
term and 3( 8mn  5 ) is the second term.
We factor out what these 2 terms have in common in this case
they have 8mn  5 in common
So, we factor out 8mn  5 and we will have ( 8mn  5 ) ( m  3
) this is the answer
The m  3 is what is left after we factored out 8mn  5
We can check the answer by multiplying ( 8mn  5 ) ( m  3 )
use the FOIL method
8m^{ 2} n  24mn  5m + 15 this answer matches the
original problem (order doesn't matter when adding)
Example 3
Factor 15x  3xy + 4y  20
We have 4 terms, so we use factoring by grouping.
The first thing we do is the group the first 2 terms together
and group the last 2 terms together. We do this by inserting
parentheses.
(15x  3xy) (4y  20)
Now we work with the groups. We factor out the GCF in the
first group and the GCF in the second group: 3x is the first GCF
and 4 is the second GCF:
3x ( 5  y ) + 4 ( y  5)
Notice that we don't have a common factor here. We have 5  y
and y  5. They almost match but not quite. What we need to do is
change the signs in the y  5 and then we'll have a match. What
we can do is to go back and factor out 4 instead of +4.
3x ( 5  y ) 4 ( y +5) 
[5  y is the same as y + 5] 
We factor out what these 2 terms have in common  in this case
they have 5  y in common
So, we factor out 5  y and we will have (5  y) (3x  4) this
is the answer
We can check the answer by multiplying (5  y) (3x  4) use
the FOIL method
15x  20  3xy + 4y this answer matches the original problem
(order doesn't matter when adding)
