Factoring Trinomials in FormFactor each trinomial.
(a)
To factor this trinomial, we use an alternative method. The goal is to find the
correct numbers to put in the blanks.
Addition signs are used, since all the signs in the polynomial indicate addition. The
first two expressions have a product of so they must be 3xand x.
The product of the two last terms must be 2, so the numbers must be 2 and 1. There
is a choice. The 2 could be placed with the 3xor with the x. Only one of these
choices will give the correct middle term, 7 x. We use the FOIL method to check
each one.
3 x 6 x2 xxWrong middle term Correct middle termTherefore, factors as (Compare with the solution
obtained by factoring by grouping on the preceding page.)
(b)
To reduce the number of trials, we note that the terms of the trinomial have
greatest common factor 1.This means that neither of its factors can have a com-
mon factor except 1.We should keep this in mind as we choose factors. We try 4 and
3 for the two first terms.
The factors of are and 1 or and 2. We try both possibilities.
8 r14 r - 2213 r+ 12 14 r- 1213 r + 22
- 2 - 2 - 1
12 r^2 - 5 r - 2 = 14 r^213 r^2
12 r^2 - 5 r- 2
3 x^2 + 7 x+ 2 13 x+ 121 x+ 22.
3 x+ 2 x= 5 x 6 x+x= 7 x13 x+ 221 x+ 12 13 x+ 121 x+ 22
3 x^2 + 7 x+ 2 = 13 x+^21 x+^2
3 x^2 ,
3 x^2 + 7 x+ 2 = 1 x+^21 x+^2
3 x^2 + 7 x+ 2
EXAMPLE 6 ax (^2) +bx+c
SECTION 6.2 Factoring Trinomials 329
Wrong: has a
factor of 2, which
cannot be correct,
since 2 is not a factor
of 12r^2 - 5 r-2.4 r- 2Wrong middle term8 r- 3 r= 5 r- 3 r
The middle term on the right is 5r, instead of the that is needed. We get by
interchanging the signs of the second terms in the factors.
3 rCorrect middle termThus, 12 r^2 - 5 r- 2 factors as 14 r+ 1213 r- 22 .(Compare with Example 5.)
- 8 r+ 3 r=- 5 r
14 r+ 1213 r - 22
- 8 r
- 5 r - 5 r
NOW TRY
EXERCISE 6
Factor 10r^2 + 19 r+6.
NOW TRY ANSWER
- 12 r+ 3215 r+ 22
NOW TRY