NOTE As shown in Example 6(b),if the terms of a polynomial have greatest com-
mon factor 1, then none of the terms of its factors can have a common factor (except 1).
Remembering this will eliminate some potential factors.
330 CHAPTER 6 Factoring
Factoring (GCF of a, b, cis 1)
Step 1 Find pairs whose product is a.Write all pairs of integer factors of
a, the coefficient of the second-degree term.
Step 2 Find pairs whose product is c.Write all pairs of integer factors of
c, the last term.
Step 3 Choose inner and outer terms.Use FOIL and various combina-
tions of the factors from Steps 1 and 2 until the necessary middle
term is found.If no such combinations exist, the trinomial is prime.
ax^2 bxcFactoring a Trinomial in Two VariablesFactor
The terms of the polynomial have greatest common factor 1. Follow the steps for
factoring a trinomial. There are many possible factors of both 18 and Try 6 and
3 for 18 and and 4 for
Wrong: common factor Wrong: common factorsSince 6 and 3 do not work as factors of 18, try 9 and 2 instead, with 3 and as
factors of
27 mxWrong: common factorsWrong middle termThe result on the right differs from the correct middle term only in sign, so inter-
change the signs of the second terms in the factors.
18 m^2 - 19 mx- 12 x 2 factors as 19 m+ 4 x 212 m- 3 x 2
27 mx+(- 8 mx)= 19 mx- 8 mx
19 m+ 3 x 212 m- 4 x 2 19 m- 4 x 212 m+ 3 x 2
- 12.
- 4
16 m- 3 x 213 m+ 4 x 2 16 m+ 4 x 213 m- 3 x 2
- 3 - 12.
- 12.
18 m^2 - 19 mx- 12 x^2.
EXAMPLE 7
Check by
multiplying.NOW TRYFactoring a Trinomial in FormFactor
While we could factor directly, it is helpful to first factor out so that the coef-
ficient of the -term is positive.
Factor out
Factor the trinomial.=- 13 x+ 221 x- 62 - 1 a=-a NOW TRY
=- 113 x+ 221 x- 62
= - 113 x^2 - 16 x- 122 - 1.
- 3 x^2 + 16 x+ 12
x^2
- 1
- 3 x^2 + 16 x+12.
EXAMPLE 8 ax (^2) +bx+c 1 a 602
NOW TRY
EXERCISE 7
Factor 12a^2 - 19 ab- 21 b^2.
NOW TRY ANSWERS
- 14 a+ 3 b 213 a- 7 b 2
NOW TRY
EXERCISE 8
Factor - 8 x^2 + 22 x-15.
- 14 x- 5212 x- 32