Beginning Algebra, 11th Edition

Factoring a Trinomial with Two Negative Terms

Factor

Find two integers whose product is and whose sum is Because the con-

stant term, -15,is negative, list pairs of integers with different signs.

- 15 - 2.

p^2 - 2 p-15.

EXAMPLE 4

370 CHAPTER 6 Factoring and Applications

NOW TRY
EXERCISE 5
Factor each trinomial if
possible.

(a)

(b)t^2 + 11 t- 24

m^2 + 5 m+ 8

Factors of 15 Sums of Factors

15,

1

5,

• 5 , 3 - 5 + 3 =- 2
  
  
  • 3 5 + 1 - 32 = 2
  
  
  
  


• 15, - 15 + 1 =- 14


• 1 15 + 1 - 12 = 14


• 
  

Sum is -2.

The required integers are and 3.

p^2 - 2 p- 15 factors as 1 p- 521 p+ 32. NOW TRY

- 5

To check, multiply

the factored form.

NOTE In Examples 1– 4,notice that we listed factors in descending order (disre-

garding their signs) when we were looking for the required pair of integers. This

helps avoid skipping the correct combination.

Some trinomials cannot be factored by using only integers. We call such trino-

mials prime polynomials.

Deciding Whether Polynomials Are Prime

Factor each trinomial if possible.

(a)

As in Example 2,both factors must be negative to give a positive product and a

negative sum. List pairs of negative integers whose product is 12, and examine the

sums.

x^2 - 5 x+ 12

EXAMPLE 5

Factors of 12 Sums of Factors

• 4, - 3 - 4 + 1 - 32 =- 7


• 6, - 2 - 6 + 1 - 22 =- 8


• 12, - 1 - 12 + 1 - 12 =- 13

None of the pairs of integers has a sum of Therefore, the trinomial

cannot be factored by using only integers.It is a prime polynomial.

(b)

There is no pair of integers whose product is 11 and whose sum is so

k^2 - 8 k+ 11 is a prime polynomial.

- 8,

k^2 - 8 k+ 11

- 5. x^2 - 5 x+ 12

Guidelines for Factoring

Find two integers whose product is cand whose sum is b.

1. Both integers must be positive if band care positive. (See Example 1.)


2. Both integers must be negative if cis positive and bis negative. (See
   Example 2.)


3. One integer must be positive and one must be negative if cis negative.
   (See Examples 3 and 4.)

x^2 bxc

NOW TRY
EXERCISE 4
Factor .x^2 - 4 x- 21

NOW TRY ANSWERS
4.

5. (a)prime (b)prime

1 x+ 321 x- 72

NOW TRY

No sum is -5.

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