Intermediate Algebra (11th edition)

Factoring by Grouping

Factor

Group the terms.

= 31 x- y 2 + a 1 - x+y 2 Factor out 3, and factor out a.

= 13 x- 3 y 2 + 1 - ax+ ay 2

3 x- 3 y- ax+ ay

3 x- 3 y- ax+ay.

EXAMPLE 6

SECTION 6.1 Greatest Common Factors and Factoring by Grouping 323

Pay close attention here.

The factors and are opposites. If we factor out instead of ain

the second group, we get the common binomial factor So we start over.

Factor out

CHECK

Multiply using the FOIL method.

= 3 x- 3 y-ax +ay ✓ Original polynomial

= 3 x- ax- 3 y+ay

1 x-y 213 - a 2

= 1 x-y 213 - a 2 x-y.

= 31 x- y 2 - a 1 x-y 2

13 x- 3 y 2 + 1 - ax +ay 2

1 x-y 2.

1 x- y 2 1 - x+y 2 - a

Be careful

with signs.

NOW TRY
EXERCISE 6
Factor.

ab- 7 a- 5 b+ 35

NOW TRY ANSWERS

6. 1 b- 721 a- 52

NOTE In Example 6, a different grouping would lead to the factored form

1 a- 321 y- x 2 .Verify by multiplying that this is also correct.

Factoring by Grouping

Factor.

Group the terms.

Now factor 6xfrom the first group, and use the identity property of multiplication to

introduce the factor 1 in the second group.

Factor each group.

Factor out

CHECK

FOIL

= 6 ax+ 12 bx+a+ 2 b ✓ Original polynomial

= 6 ax+ a+ 12 bx + 2 b

1 a+ 2 b 216 x+ 12

= 1 a+ 2 b 216 x+ 12 a+ 2 b.

= 6 x 1 a+ 2 b 2 + 11 a+ 2 b 2

= 16 ax+ 12 bx 2 + 1 a+ 2 b 2

6 ax + 12 bx+ a+ 2 b

6 ax+ 12 bx +a+ 2 b

EXAMPLE 7

NOW TRY

Factoring by Grouping

Step 1 Group terms.Collect the terms into groups so that each group has

a common factor.

Step 2 Factor within the groups.Factor out the common factor in each

group.

Step 3 Factor the entire polynomial.If each group now has a common

factor, factor it out. If not, try a different grouping.

Always check the factored form by multiplying.

Remember to

write the 1.

NOW TRY

NOW TRY
EXERCISE 7
Factor.

3 ax- 6 xy-a+ 2 y

7. 1 a- 2 y 213 x- 12