Beginning Algebra, 11th Edition

NOTE In Example 4,we might also realize that our initial attempt to factor

as cannotbe correct, since the terms of

have a common factor of 3, while those of the original polynomial do not.

6 x^2 - 11 x+ 3 16 x- 321 x- 12 6 x- 3

Now try the numbers 2 and 3 as factors of 6. Because of the common factor 2 in

the product will not work, so we try

Correct

Add.

Thus, 2 x^2 + 7 x+ 6 factors as 12 x+ 321 x+ 22.

7 x

4 x

3 x

12 x+ 321 x+ 22 = 2 x^2 + 7 x+ 6

2 x+ 2, 12 x+ 221 x+ 32 12 x+ 321 x+ 22.

376 CHAPTER 6 Factoring and Applications

Incorrect Incorrect Correct

13 p Add. 41 p Add. 14 p Add.

8 p p 4 p

5 p 40 p 10 p

18 p+ 521 p+ 12 1 p+ 5218 p+ 12 14 p+ 5212 p+ 12

Since the combination on the right produces 14p, the correct middle term,

factors as

CHECK Multiply to obtain 14 p+ 5212 p+ 12 8 p^2 + 14 p+ 5 ✓

8 p^2 + 14 p+ 5 14 p+ 5212 p+ 12.

Factoring a Trinomial with a Negative Middle Term

by Using FOIL

Factor

Since 3 has only 1 and 3 or and as factors, it is better here to begin by

factoring 3. The last (constant) term of the trinomial is positive and

the middle term has a negative coefficient, so we consider only negative factors. We

need two negative factors, because the productof two negative factors is positive and

their sumis negative, as required. Try and as factors of 3.

The factors of may be either 6xand xor 2xand 3x.

Incorrect Correct

Add. Add.

The factors 2xand 3xproduce the correct middle term.

6 x^2 - 11 x+ 3 factors as 12 x- 3213 x- 12. NOW TRY

- 11 x,

• 9 x - 11 x


• 6 x - 2 x


• 3 x - 9 x

16 x- 321 x- 12 12 x- 3213 x- 12

6 x^2

1 - 321 - 12

- 3 - 1

6 x^2 - 11 x+ 3

- 1 - 3

6 x^2 - 11 x+ 3.

EXAMPLE 4

Checkby multiplying.

NOW TRY
EXERCISE 3
Factor. 8 y^2 + 22 y+ 5

NOW TRY
EXERCISE 4
Factor. 10 x^2 - 9 x+ 2

NOW TRY ANSWERS
3.

4. 15 x- 2212 x- 12

14 y+ 1212 y+ 52

Factoring a Trinomial with All Positive Terms by Using FOIL

Factor

The number 8 has several possible pairs of factors, but 5 has only 1 and 5 or

and so begin by considering the factors of 5. Ignore the negative factors, since

all coefficients in the trinomial are positive. The factors will have this form.

The possible pairs of factors of are 8pand p, or 4pand 2p. Try various combi-

nations, checking in each case to see if the middle term is 14p.

8 p^2

1 + 521 + 12

- 5,

- 1

8 p^2 + 14 p+5.

EXAMPLE 3

NOW TRY

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