In-class Handout, Sections 6.2 and 6.3 (Factoring Trinomials), July 3rd, 2012
Section 6.2 Factoring Trinomials I:
𝒙𝟐 + 𝒃𝒙 + 𝒄
(𝒂 = 𝟏), p. 380
Some trinomials can be factored into the product of two binomials. Factoring these trinomials requires answering the
following question.

Is there are pair of integers that multiply to 𝒄 and sum to 𝒃?
o If “no”, then the trinomial is prime (see second example below).
o If “yes”, then substitute those two integers into two binomials as shown below.
Example, factor 𝒙𝟐 − 𝟗𝒙 − 𝟑𝟔
𝒂=𝟏
𝒃 = −𝟗
𝒄 = −𝟑𝟔
𝑥 2 − 9𝑥 − 36


Is there a pair of integers that multiply to −36 and sum to −9?
o Yes, 3 and −12.
𝑥 2 − 9𝑥 − 36 = (𝑥 + 3)(𝑥 − 12)
Answer


Check your result by FOILing the answer.
See pages 381-383 for additional examples.
Not all trinomials can be factored. If trinomial cannot be factored, it is called prime.
Example, factor 𝒙𝟐 − 𝟖𝒙 − 𝟑𝟔
𝒂=𝟏
𝒃 = −𝟖
𝒄 = −𝟑𝟔
𝑥 2 − 8𝑥 − 36

Is there a pair of integers that multiply to −36 and sum to −8?
o No. (Look at the factor pairs listed above right.)
o The trinomial 𝑥 2 − 8𝑥 − 36 is prime and cannot be factored.
Some Factor
Pairs of −36
Their
Sums
1 − 36
−35
2 − 18
−16
3 − 12
−9
4
−9
−5
6
−6
0
Section 6.3 Factoring Trinomials II:
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
(𝒂 > 𝟏), p. 387
Trinomials with a leading coefficient other than an implied one may take additional effort to factor. Numerous
methods exist. The two taught in the textbook are the “𝒂𝒄-method with grouping” and “FOIL in reverse.” Names of
common methods:


Formal Methods That Always Work
o 𝒂𝒄-method with grouping (p. 387)
o 𝒂𝒄-method without grouping (in class)
o 𝒂𝒄-method with fake factoring (SPA has a worksheet)
Trial-and-Error Methods
o “FOIL in reverse” (p. 390)
o X-factoring (a visual trial-and-error method)
𝒂𝒄-method with grouping (p. 387)

Is there are pair of integers that multiply to 𝒂𝒄 and sum to 𝒃?
o If “no”, then the trinomial is prime.
o If “yes”, then change the trinomial to a 4-term polynomial and then factor by grouping (see below).
Example, factor 𝟐𝒙𝟐 + 𝒙 − 𝟐𝟖
Multiply 𝑎 and 𝑐
𝒂=𝟐
𝒃=𝟏
𝒄 = −𝟐𝟖
2𝑥 2 + 𝑥 − 28


Is there a pair of integers that multiply to −56 and sum to 1?
o Yes, −7 and 8.
2𝑥 2 + 𝑥 − 28
=
=
=
2𝑥 2 − 7𝑥 + 8𝑥 − 28
𝑥(2𝑥 − 7) + 4(2𝑥 − 7)
(2𝑥 − 7)(𝑥 + 4)
𝒂𝒄 = −𝟓𝟔
Some Factor
Pairs of −56
Their
Sums
1 − 56
−55
2 − 28
−26
4 − 14
−10
7
−1
−7
Answer
−8
8
1