Factoring Trinomials in FormFactor each trinomial.
(a)x^2 + 2 x- 35
EXAMPLE 1 x (^2) +bx+c
SECTION 6.2 Factoring Trinomials 327
Step 1 Find pairs of integers
whose product is
51 - 72
71 - 52
351 - 12
- 35112
- 35.
Step 2 Write sums of those
integers.
5 + 1 - 72 =- 2
7 + 1 - 52 = 2
35 + 1 - 12 = 34
- 35 + 1 =- 34
Coefficient of the
middle termThe integers 7 and have the necessary product and sum, so
x^2 + 2 x- 35 factors as 1 x+ 721 x- 52.
- 5
Multiply to
check.(b)
Look for two integers with a product of 12 and a sum of 8. Of all pairs having a
product of 12, only the pair 6 and 2 has a sum of 8.
factors as
Because of the commutative property, it would be equally correct to write
Check by using FOIL to multiply the factored form. NOW TRY
Recognizing a Prime PolynomialFactor
Look for two integers whose product is 7 and whose sum is 6. Only two pairs of
integers, 7 and 1 and and give a product of 7. Neither of these pairs has a sum
of 6, so cannot be factored with integer coefficients and is prime.
NOW TRYFactoring a Trinomial in Two VariablesFactor
This trinomial is in the form x^2 + bx+ c,where b= 6 aand c=- 16 a^2.
x^2 + 6 ax- 16 a^2.
EXAMPLE 3
m^2 + 6 m+ 7
- 7 - 1,
m^2 + 6 m+ 7.
EXAMPLE 2
1 r+ 221 r+ 62.
r^2 + 8 r+ 12 1 r+ 621 r+ 22.
r^2 + 8 r+ 12
NOW TRY
EXERCISE 1
Factor each trinomial.
(a)
(b)w^2 + 12 w+ 32
t^2 - t- 30NOW TRY ANSWERS
- (a)
(b) 1 w+ 421 w+ 82
1 t- 621 t+ 52NOW TRY
EXERCISE 2
Factor m^2 + 12 m-11.
- prime
NOW TRY
EXERCISE 3
Factor a^2 +ab- 20 b^2.
- 1 a+ 5 b 21 a- 4 b 2
Step 1 Find pairs of expressions
whose product is
- 4 a 14 a 2
- 8 a 12 a 2
8 a 1 - 2 a 2
- 16 a 1 a 2
16 a 1 - a 2
- 16 a^2.
Step 2 Write sums of the pairs of
expressions from Step 1,
looking for a sum of 6 a.
- 4 a+ 4 a= 0
- 8 a+ 2 a=- 6 a
8 a+ 1 - 2 a 2 = 6 a
- 16 a+ a=- 15 a
16 a+ 1 - a 2 = 15 a
The expressions 8aand have the necessary product and sum, so
factors as
CHECK
FOIL