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)
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