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Factoring Polynomials by Grouping

6.4.4 Factoring Polynomials by Grouping

6.4.4 • Factoring Polynomials by Grouping

Activity

If there is no common term in all of the terms of a polynomial, it may be possible to factor by grouping. To do this, separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find that a common factor emerges from both parts.

Let’s look at an example. Factor the following polynomial.

m s − 4 m + 7 s − 28 m s − 4 m + 7 s − 28

Compare your answer:

Step 1 - Group terms with common factors.

Is there a GCF of all four terms? No, so let’s separate the first two terms from the second two.

m s − 4 m + 7 s − 28 m s − 4 m + 7 s − 28

The expression ms minus 4m plus 7s minus 28 with red brackets grouping ms minus 4m and 7s minus 28 separately underneath.

Step 2 - Factor out the common factor in each group.

Factor the GCF from the first two terms. Factor the GCF from the second two terms.

m ( s − 4 ) + 7 s − 28 m ( s − 4 ) + 7 s − 28

m ( s − 4 ) + 7 ( s − 4 ) m ( s − 4 ) + 7 ( s − 4 )

Step 3 - Factor the common factor from the expression.

Notice that each term has a common factor of ( s − 4 ) ( s − 4 ) . Factor out the common factor.

m ( s − 4 ) + 7 ( s − 4 ) m ( s − 4 ) + 7 ( s − 4 )

( s − 4 ) ( m + 7 ) ( s − 4 ) ( m + 7 )

Step 4 - Check.

Multiply ( s − 4 ) ( m + 7 ) ( s − 4 ) ( m + 7 ) . Is the product the original expression?

( s − 4 ) ( m + 7 ) ( s − 4 ) ( m + 7 )

m s + 7 s − 4 m − 28 m s + 7 s − 4 m − 28

m s − 4 m + 7 s − 28 ✓ m s − 4 m + 7 s − 28 ✓

Video: Factoring by Grouping

Watch the following video to learn more about how to factor by grouping.

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Work on the following exercises individually.

Factor by grouping:

1. x y + 7 y + 8 x + 56 x y + 7 y + 8 x + 56

Compare your answer:

( x + 7 ) ( y + 8 ) ( x + 7 ) ( y + 8 )

2. x 2 + 2 x − 5 x − 10 x 2 + 2 x − 5 x − 10

You can use the following video if you need some help

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Compare your answer:

( x − 5 ) ( x + 2 ) ( x − 5 ) ( x + 2 )

3. 20 x 2 − 16 x − 15 x + 12 20 x 2 − 16 x − 15 x + 12

Compare your answer:

( 5 x − 4 ) ( 4 x − 3 ) ( 5 x − 4 ) ( 4 x − 3 )

4. x 2 + 4 x − 7 x − 28 x 2 + 4 x − 7 x − 28

Compare your answer:

( x + 4 ) ( x − 7 ) ( x + 4 ) ( x − 7 )

5. 2 x 2 − 14 x − 5 x + 35 2 x 2 − 14 x − 5 x + 35

Compare your answer:

( x − 7 ) ( 2 x − 5 ) ( x − 7 ) ( 2 x − 5 )

6. x y − 6 x − 4 y + 24 x y − 6 x − 4 y + 24

Compare your answer:

( y − 6 ) ( x − 4 ) ( y − 6 ) ( x − 4 )

7. 9 x 2 + 12 x − 15 x − 20 9 x 2 + 12 x − 15 x − 20

Compare your answer:

( 3 x − 5 ) ( 3 x + 4 ) ( 3 x − 5 ) ( 3 x + 4 )

8. x y − 9 x + 2 y − 18 x y − 9 x + 2 y − 18

Compare your answer:

( x + 2 ) ( y − 9 ) ( x + 2 ) ( y − 9 )

Self Check

Factor by grouping: 42 m 2 − 18 m − 35 m + 15 .
  1. ( 7 m − 5 ) ( 6 m − 3 )
  2. ( 7 m + 3 ) ( 6 m + 5 )
  3. ( 7 m − 3 ) ( 6 m − 5 )
  4. ( 14 m − 3 ) ( 3 m − 5 )

Additional Resources

Factoring Polynomials by Grouping

Sometimes there is no common factor of all the terms of a polynomial. When there are four terms, we separate the polynomial into two parts with two terms in each part. Then we look for the GCF in each part. If the polynomial can be factored, you will find that a common factor emerges from both parts. Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.

Example 1

Factor by grouping: x y + 3 y + 2 x + 6 x y + 3 y + 2 x + 6 .

Step 1 - Group terms with common factors.

Is there a greatest common factor of all four terms?

x y + 3 y + 2 x + 6 x y + 3 y + 2 x + 6

No, so let’s separate the first two terms from the second two.

Mathematical expression xy + 3y + 2x + 6 with two teal brackets below, one grouping xy + 3y and the other grouping 2x + 6.

Step 2 - Factor out the common factor in each group.

Factor the GCF from the first two terms.

Mathematical expression: y times the quantity x plus 3, plus 2x plus 6; a teal bracket underneath highlights the terms 2x plus 6.

Factor the GCF from the second two terms.

y ( x + 3 ) + 2 ( x + 3 ) y ( x + 3 ) + 2 ( x + 3 )

Step 3 - Factor the common factor from the expression.

Notice that each term has a common factor of ( x + 3 ) ( x + 3 ) .

y ( x + 3 ) + 2 ( x + 3 ) y ( x + 3 ) + 2 ( x + 3 )

Factor out the common factor.

( x + 3 ) ( y + 2 ) ( x + 3 ) ( y + 2 )

Step 4 - Check.

Multiply ( x + 3 ) ( y + 2 ) ( x + 3 ) ( y + 2 ) . Is the product the original expression?

( x + 3 ) ( y + 2 ) ( x + 3 ) ( y + 2 )

x y + 2 x + 3 y + 6 x y + 2 x + 3 y + 6

x y + 3 y + 2 x + 6 ✓ x y + 3 y + 2 x + 6 ✓

How to factor by grouping:

Step 1 - Group terms with common factors.

Step 2 - Factor out the common factor in each group.

Step 3 - Factor the common factor from the expression.

Step 4 - Check by multiplying the factors.

Example 2

Factor by grouping: x 2 + 3 x − 2 x − 6 x 2 + 3 x − 2 x − 6 .

Step 1 - There is no GCF in all four terms.

x 2 + 3 x − 2 x − 6 x 2 + 3 x − 2 x − 6

Step 2 - Separate into two parts.

x 2 + 3 x − 2 x − 6 x 2 + 3 x − 2 x − 6

Step 3 - Factor the GCF from both parts. Be careful with the signs when factoring the GCF from the last two terms.

x ( x + 3 ) − 2 ( x + 3 ) x ( x + 3 ) − 2 ( x + 3 )

Step 4 - Factor out the common factor.

( x + 3 ) ( x − 2 ) ( x + 3 ) ( x − 2 )

Step 5 - Check on your own by multiplying.

Try it

Factoring Polynomials by Grouping

Factor by grouping:

1. x y + 8 y + 3 x + 24 x y + 8 y + 3 x + 24

2. 6 x 2 − 3 x − 4 x + 2 6 x 2 − 3 x − 4 x + 2

Here is how to factor polynomials by grouping.

1. Step 1 - There is no GCF in all four terms.

x y + 8 y + 3 x + 24 x y + 8 y + 3 x + 24

Step 2 - Separate into two parts.

x y + 8 y + 3 x + 24 x y + 8 y + 3 x + 24

Step 3 - Factor the GCF from both parts.

y ( x + 8 ) + 3 ( x + 8 ) y ( x + 8 ) + 3 ( x + 8 )

Step 4 - Factor out the common factor.

( x + 8 ) ( y + 3 ) ( x + 8 ) ( y + 3 )

Step 5 - Check on your own by multiplying.

2. Step 1 - There is no GCF in all four terms.

6 x 2 − 3 x − 4 x + 2 6 x 2 − 3 x − 4 x + 2

Step 2 - Separate into two parts.

6 x 2 − 3 x − 4 x + 2 6 x 2 − 3 x − 4 x + 2

Step 3 - Factor the GCF from both parts.

3 x ( 2 x − 1 ) − 2 ( 2 x − 1 ) 3 x ( 2 x − 1 ) − 2 ( 2 x − 1 )

Step 4 - Factor out the common factor.

( 2 x − 1 ) ( 3 x − 2 ) ( 2 x − 1 ) ( 3 x − 2 )

Step 5 - Check on your own by multiplying.