Factoring by Grouping

Objective: After completing this section, students should be able to factor polynomials by grouping.

Steps for factoring by grouping:

1. A polynomial must have 4 terms to factor by grouping.

2. We factor the first two terms and the second two terms separately. Use the rules for GCF to factor these.

3 2. 2 2x x xex

3 2 2 2xx x 3 2 2

The GCF of

is .x x x

21 2x x

The GCF of

2 2 is 2.x

2

3. Finally, we factor out the "common factor" from both terms.

This means we write the 1 term in front and the 2 terms

left over, +2 , in a separate set of parentheses.

x

x

2 1x x 2 1x

Examples:

3 21. 6 9 4 6x x x 3 2 4 66 9 xx x

3 2 2

The GCF of

6 9 is 3 .x x x

The GCF of

4 6 is 2.x 23 2 3x x 2 2 3x

These two terms must be the same. 22 3 3 2x x

3 22. 1x x x 3 2 1xx x

3 2 2

The GCF of

is .x x x

The GCF of

1 is 1.x 2 1x x 1 1x

These two terms must be the same. 21 1x x

Examples:

3 23. 2 2x x x 3 2 22 xx x

3 2 2

The GCF of

2 is .x x x

The GCF of

2 is 1.x 2 2x x 1 2x

These two terms must be the same. 22 1x x

You must always check to see if the expression is factored completely. This expression can still be factored using the rules for difference of two squares. (see 6.2)

22 1x x

2 1 1x x x

This is a difference of two squares.

Examples:

2 2 2 24. x y ay ab bx 2 2 2 2x y ay ab bx

2 2 2 2

The GCF of

is .x y ay y 2

The GCF of

is .ab bx b 2 2y x a 2b a x

These two terms must be the same.

You can rearrange the terms so that they are the same.

2 2y b x a

3 25. 2 2x x x 3 2 2 2xx x

3 2 2

The GCF of

is .x x x

The GCF of

2 2 is 2.x 2 1x x 2 1x

These two terms must be the same.

But they are not the same. So this polynomial is not factorable.

Not Factorable

Try These:Factor by grouping.

3 2

3 2

3 2

2

a. 8 2 12 3

b. 4 6 6 9

c. 1

d. 3 6 5 10

x x x

x x x

x x x

a b a ab

Solutions: If you did not get these answers, click the green button next to the solution to see it worked out.

2

2

a. 4 1 2 3

b. 2 3 2 3

c. 1 1 1

d. 2 3 5

x x

x x

x x x

a b a

BACK

3 2a. 8 2 12 3x x x

3 2

2

2

8 2 12 3

3 4 12 4 1

4 1 2 3

x x x

xx x

x x

3 2 2

The GCF of

8 2 is 2 .x x x

The GCF of

12 3 is 3.x

BACK

3 2b. 4 6 6 9x x x

3 2

2

2

4 6 6 9

3 2 32 2 3

2 3 2 3

x x x

xx x

x x

3 2 2

The GCF of

4 6 is 2 .x x x

The GCF of

6 9 is 3.x

When you factor a negative out of a positive, you will get a negative.

BACK

3 2c. 1x x x

3 2

2

2

1

1 11

1 1

1 1 1

x x x

xx x

x x

x x x

3 2 2

The GCF of

is .x x x

The GCF of

1 is 1.x

Now factor the difference of squares.

BACK

2d. 3 6 5 10a b a ab

23 6 5 10

3 2 5 2

2 3 5

a b a ab

a b a a b

a b a

The GCF of

3 6 is 3.a b 2

The GCF of

5 10 is 5 .a ab a