Intermediate Algebra (11th edition)

Factoring Out a Negative Common Factor

Factor in two ways.

First, acould be used as the common factor.

Factor out a.

Because of the leading negative sign, could also be used as the common factor.

Factor out

Sometimes there may be a reason to prefer one of these forms, but either is correct.

NOW TRY

NOTE The answer section in this book will usually give the factored form where the

common factor has a positive coefficient.

OBJECTIVE 2 Factor by grouping.Sometimes the individual termsof a poly-

nomial have a greatest common factor of 1, but it still may be possible to factor the

polynomial by using a process called factoring by grouping.We usually factor by

grouping when a polynomial has more than three terms.

= -a 1 a^2 - 3 a+ 52

= -a 1 a^22 + 1 - a 21 - 3 a 2 + 1 - a 2152 - a.

- a^3 + 3 a^2 - 5 a

- a

= a 1 - a^2 + 3 a- 52

= a 1 - a^22 +a 13 a 2 + a 1 - 52

- a^3 + 3 a^2 - 5 a

- a^3 + 3 a^2 - 5 a

EXAMPLE 4

322 CHAPTER 6 Factoring

NOW TRY
EXERCISE 4
Factor in
two ways.

• 4 y^5 - 3 y^3 + 8 y

NOW TRY ANSWERS
4.

• y 14 y^4 + 3 y^2 - 82

y 1 - 4 y^4 - 3 y^2 + 82 ;

5. 1 m-n 213 +x 2

NOW TRY
EXERCISE 5
Factor.

3 m- 3 n+xm-xn

NOW TRY

Factoring by Grouping

Factor

Groupthe terms as follows.

Terms with common factor a Terms with common factor b

Then factor as and factor as.

Group the terms.

Factor each group.

The common factor is.

CHECK

Multiply using the FOIL method.

Commutative property

= ax- ay+bx -by ✓ Original polynomial

= ax+ bx-ay -by

= xa+ xb-ya -yb

1 x-y 21 a+ b 2

= 1 x- y 21 a+b 2 x-y

= a 1 x-y 2 + b 1 x-y 2

= 1 ax-ay 2 + 1 bx- by 2

ax -ay+ bx- by

ax- ay a 1 x-y 2 bx- by b 1 x-y 2

1 ax- ay 2 + 1 bx- by 2

ax -ay+ bx-by.

EXAMPLE 5