Beginning Algebra, 11th Edition

SECTION 6.3 More on Factoring Trinomials^377

Factoring a Trinomial with a Negative Constant Term

by Using FOIL

Factor

The integer 8 has several possible pairs of factors, as does Since the constant

term is negative, one positive factor and one negative factor of are needed. Since

the coefficient of the middle term is relatively small, it is wise to avoid large factors

such as 8 or 9. We try 4xand 2xas factors of 8 x^2 , and 3 and - 3 as factors of -9.

- 9

- 9.

8 x^2 + 6 x-9.

EXAMPLE 5

Factoring a Trinomial with Two Variables

Factor

There are several pairs of factors of including

There are also many pairs of factors of including

Once again, since the coefficient of the desired middle term is relatively small, avoid

the larger factors. Try the factors 6aand 2a, and 4band

This cannot be correct, since there is a factor of 2 in , while 2 is not a factor

of the given trinomial. Try 3aand 4awith 4band

Incorrect

Here the middle term is abrather than , so we interchange the signs of the last

two terms in the factors.

12 a^2 - ab- 20 b^2 factors as 13 a- 4 b 214 a+ 5 b 2.

- ab

= 12 a^2 + ab- 20 b^2

13 a+ 4 b 214 a- 5 b 2

- 5 b.

6 a+ 4 b

16 a+ 4 b 212 a- 5 b 2

- 5 b.

4 b and - 5 b, and - 4 b and 5b.

20 b and -b, - 20 b and b, 10 b and - 2 b, - 10 b and 2b,

- 20 b^2 ,

12 a and a, 6 a and 2a, and 3 a and 4a.

12 a^2 ,

12 a^2 - ab- 20 b^2.

EXAMPLE 6

NOW TRY
EXERCISE 5
Factor. 10 a^2 + 31 a- 14

NOW TRY
EXERCISE 6
Factor. 8 z^2 + 2 wz- 15 w^2

NOW TRY ANSWERS
5.

6. 14 z- 5 w 212 z+ 3 w 2

15 a- 2212 a+ 72

Factoring Trinomials with Common Factors

Factor each trinomial.

(a)

Factor out the greatest common factor, 5y.

To factor try 3yand yas factors of , and 2 and 3 as factors of 6.

= 3 y^2 + 11 y+ 6 Correct

13 y+ 221 y+ 32

3 y^2 + 11 y+ 6, 3 y^2

= 5 y 13 y^2 + 11 y+ 62

15 y^3 + 55 y^2 + 30 y

EXAMPLE 7

Incorrect Correct

• 6 x Add. 6 x Add.


• 12 x 12 x

6 x - 6 x

14 x+ 3212 x- 32 14 x- 3212 x+ 32

The combination on the right produces the correct middle term.

8 x^2 + 6 x- 9 factors as 14 x- 3212 x+ 32. NOW TRY

Interchange 3 and 3,

since only the sign of the

middle term is incorrect.

• 
  

Checkby

multiplying.

NOW TRY