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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Factoring By Grouping

After studying this lesson, you will be able to:

• Factor by grouping.

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)