8.8 Factoring by Grouping

Factoring by grouping

USE WHEN THERE ARE 4 TERMS IN THE POLYNOMIAL.• Polynomials with four or more terms like 3xy – 21y + 5x – 35,

can sometimes be factored by grouping terms of the polynomials. The key is to group the terms into binomials that can be factored using the distributive property.

• Then use the distributive property again with a binomial as the common factor.

Factor 3xy – 21y + 5x – 35

)355()213(355213 xyxyxyxy

)7)(53(

)7(5)7(3

xy

xxy

355213

)7(5)(5)7(3)(3)7)(53(

xyxy

xyxyxy

Group terms that have common monomial factor.

Factor. Notice that (x – 7) is a common factor.

Distributive property.

Check by using FOIL.

Factor 152458 2 mnmnm

152458 2 mnmnm

)58)(3(

)58)(3()58(

)1524()58( 2

mnm

mnmnm

mnmnm

152458

)5)(3()8)(3()5()8()58)(3(2

mnmnm

mnmmnmmnm

Group terms that have common monomial factor.

Factor. Notice that (8mn - 5) is a common factor.

Distributive property.

Check by using FOIL.

Factor acabbca 422 2

acabbca 422 2

)2)(2(

)2(2)2(

)24()2( 2

baca

bacbaa

bcacaba

acabbca 422 2

)2)(2(

)2()2(2

)2()42( 2

caba

cabcaa

bcabaca

acabbca 422 2

acabbca

bcbaaca

cbabcaaa

422

242

)2()()2(2)(2

2

2

FOIL AND CHECK

Note:

• Recognizing binomials that are additive inverses is often helpful in factoring. For example, the binomials 3 – a and a – 3 are additive inverses since the sum of 3 – a and a – 3 is 0. Thus, 3 – a and –a +3 are equivalent. What is the additive inverse of 5 – y?

-y + 5

Factor:

204315 yxyx

204315 yxyx

)5)(43(

)5(4)5(3

)5(4)5(3

)204()315(

yx

yyx

yyx

yxyx

)5)(43(: yxCheck

FOIL AND CHECK

204315

204153

)5(4)(4)5)(3())(3(

yxyx

yxxy

yxyx

(5-y) and (y-5) are additive inverses.

(5-y)=(-1)(y-5)