Beginning Algebra, 11th Edition

Factoring a Trinomial with a Common Factor by Grouping

Factor

Factor out the greatest common factor,

To factor find two integers whose product is

and whose sum is Factoring 210 into prime factors helps find these integers.

Combine the prime factors of into pairs in different ways,

using one positive and one negative (to get ). The factors 6 and have the

correct sum,.

Group the terms.

Factor each group.

Factor out.

= 2 x^317 x+ 3212 x- 52 NOW TRY

= 2 x^3317 x+ 3212 x- 524 7 x+ 3

= 2 x^332 x 17 x+ 32 - 517 x+ 324

= 2 x^33114 x^2 + 6 x 2 + 1 - 35 x- 1524

= 2 x^3114 x^2 + 6 x- 35 x- 152 - 29 x= 6 x- 35 x

= 2 x^3114 x^2 - 29 x- 152

28 x^5 - 58 x^4 - 30 x^3

- 29

- 210 - 35

210 = 2 # 3 # 5 # 7

210 = 2 # 3 # 5 # 7

- 29.

14 x^2 - 29 x-15, 141 - 152 =- 210

= 2 x^3114 x^2 - 29 x- 152 2 x^3.

28 x^5 - 58 x^4 - 30 x^3

28 x^5 - 58 x^4 - 30 x^3.

EXAMPLE 2

SECTION 6.3 More on Factoring Trinomials 375

NOW TRY
EXERCISE 2
Factor. 15 z^6 + 18 z^5 - 24 z^4

OBJECTIVE 2 Factor trinomials by using the FOIL method. There is an al-

ternative method of factoring trinomials that uses trial and error.

To factor (the trinomial factored at the beginning of this section)

by trial and error, we use the FOIL method in reverse. We want to write

as the product of two binomials.

The product of the two first terms of the binomials is The possible factors of

are 2xand xor and Since all terms of the trinomial are positive, we consider

only positive factors. Thus, we have the following.

The product of the two last terms, 6, can be factored as or

Try each pair to find the pair that gives the correct middle term, 7x.

Incorrect Incorrect

Add. Add.

Since the binomial has a common factor of 2, while

has no common factor other than 1. The product

cannot be correct.

2 x^2 + 7 x+ 6 12 x+ 621 x+ 12

2 x+ 6 = 21 x+ 32 , 2 x+ 6

13 x 8 x

12 x 2 x

x 6 x

12 x+ 121 x+ 62 12 x+ 621 x+ 12

1 #6, 6 #1, 2 #3, 3 #2.

= 12 x 21 x 2

2 x^2 + 7 x+ 6

- 2 x -x.

2 x^2. 2 x^2

= 1 21 2

2 x^2 + 7 x+ 6

2 x^2 + 7 x+ 6

2 x^2 + 7 x+ 6

Remember the

common factor.

Checkby

multiplying.

NOW TRY ANSWER

2. 3 z^415 z- 421 z+ 22

NOTE If the terms of the original polynomial have greatest common factor 1, then

each factor of that polynomial will also have terms with GCF 1.