Intermediate Algebra (11th edition)

Factoring a Trinomial with a Common Factor

Factor

Factor out the GCF, 16y.

To factor look for two integers whose product is and whose sum is

The necessary integers are and 1.

= 16 y 1 y- 321 y+ 12 Factor the trinomial.

- 2. - 3

y^2 - 2 y-3, - 3

= 16 y 1 y^2 - 2 y- 32

16 y^3 - 32 y^2 - 48 y

16 y^3 - 32 y^2 - 48 y.

EXAMPLE 4

328 CHAPTER 6 Factoring

NOW TRY
EXERCISE 4
Factor 5w^3 - 40 w^2 + 60 w.

NOW TRY ANSWERS

4. 5 w 1 w- 621 w- 22

Remember to include

the GCF, 16y. NOW TRY

CAUTION When factoring, always look for a common factor first. Remember

to write the common factor as part of the answer.

OBJECTIVE 2 Factor trinomials when the coefficient of the quadratic term

is not 1.We can use a generalization of the method shown in Objective 1to factor

a trinomial of the form where To factor for

example, we first identify the values a, b, and c.

,so

The product acis so we must find integers having a product of 6 and a

sum of 7 (since the middle term has coefficient 7 ). The necessary integers are 1 and

6, so we write 7xas or.

Group the terms.

Factor by grouping.

= 13 x+ 121 x+ 22 Factor out the common factor.

=x 13 x+ 12 + 213 x+ 12

= 13 x^2 +x 2 + 16 x+ 22

x+ 6 x= 7 x

= 3 x^2 + x+ 6 x+ 2

3 x^2 + 7 x+ 2

1 x+ 6 x, x+ 6 x

3 # 2 =6,

3 x^2 + 7 x+ 2 a= 3, b=7, c= 2

ax^2 +bx+c

ax^2 + bx+ c, aZ 1. 3 x^2 + 7 x+ 2,

⎧⎪⎨⎪⎩

Check by

multiplying.

Factoring a Trinomial in Form

Factor

Since and the product acis The two integers

whose product is and whose sum bis , are 3 and.

Write as

Factor by grouping.

= 14 r + 1213 r- 22 Factor out the common factor.

= 3 r 14 r+ 12 - 214 r+ 12

= 12 r^2 + 3 r- 8 r- 2 - 5 r 3 r- 8 r.

12 r^2 - 5 r- 2

- 24 - 5 - 8

a=12,b=-5, c=-2, -24.

12 r^2 - 5 r- 2.

EXAMPLE 5 ax (^2) +bx+c
Check by
multiplying. NOW TRY

OBJECTIVE 3 Use an alternative method for factoring trinomials.This

method involves trying repeated combinations and using FOIL.

NOW TRY
EXERCISE 5
Factor 8y^2 - 10 y-3.

5. 12 y- 3214 y+ 12