Factoring a Trinomial with All Positive TermsFactor
Look for two integers whose product is 14 and whose sum is 9. List pairs of
integers whose product is 14, and examine the sums. Again, only positive integers are
needed because all signs in m^2 + 9 m+ 14 are positive.
m^2 + 9 m+ 14.
EXAMPLE 1
SECTION 6.2 Factoring Trinomials 369
NOW TRY
EXERCISE 1
Factor .p^2 + 7 p+ 10
NOW TRY
EXERCISE 2
Factor .t^2 - 9 t+ 18
Sum is 9.Factors of 14 Sums of Factors
14, 1
7 , 2 7 + 2 = 914 + 1 = 15From the list, 7 and 2 are the required integers, since and
CHECK
FOIL=m^2 + 9 m+ 14 ✓ Original polynomial NOW TRY
=m^2 + 2 m+ 7 m+ 14
1 m+ 721 m+ 22
m^2 + 9 m+ 14 factors as 1 m+ 721 m+ 22.
7 # 2 = 14 7 + 2 = 9.
Factoring a Trinomial with a Negative Middle TermFactor
We must find two integers whose product is 20 and whose sum is Since the
numbers we are looking for have a positive productand a negative sum,we consider
only pairs of negative integers.
- 9.
x^2 - 9 x+ 20.
EXAMPLE 2
Factors of 20 Sums of Factors- 5 , - 4 - 5 + 1 - 42 =- 9
- 10, - 2 - 10 + 1 - 22 =- 12
- 20, - 1 - 20 + 1 - 12 =- 21
Sum is -9.The required integers are and
factors as
CHECK
FOIL=x^2 - 9 x+ 20 ✓ Original polynomial NOW TRY
=x^2 - 4 x- 5 x+ 20
1 x- 521 x- 42
x^2 - 9 x+ 20 1 x- 521 x- 42.
- 5 - 4.
The order of the factors
does not matter.Factoring a Trinomial with a Negative Last (Constant) TermFactor
We must find two integers whose product is and whose sum is 1 (since the
coefficient of x, or 1x, is 1). To get a negative product,the pairs of integers must have
different signs.
- 6
x^2 +x-6.EXAMPLE 3
Factors of 6 Sums of Factors3 , - 2 3 + 1 - 22 = 1- 6, 1 - 6 + 1 =- 5
6, - 1 6 + 1 - 12 = 5Sum is 1.The required integers are 3 and
x^2 +x- 6 factors as 1 x+ 321 x- 22. NOW TRY
- 2.
NOW TRY
EXERCISE 3
Factor .x^2 +x- 42
NOW TRY ANSWERS
1 x+ 721 x- 62
1 t- 321 t- 621 p+ 221 p+ 52is also correct.1 m+ 221 m+ 72Once we find the
required pair, we can
stop listing factors.To check, multiply
the factored form.