College Algebra & Trigonometry, 2018a

1.2. FACTORING 19
In building the charts of factor pairs in the previous two problems, nothing more
difficult than dividing the constant term by the numbers 1, 2, 3, 4, 5, 6, ... and so
on can help you to find the full list of factor pairs. If you don’t get a whole number
when dividing - for instance 96 ÷ 5=19.2, then this number is not included
in the list of factor pairs.
Trinomial Factoring (a ≠1)
If the value of a is not 1, this means that, if the trinomial is factorable, at least one
of its binomial factors also has a coefficient other than 1. For instance:
(2x +7)(x − 3) = 2x 2 +1x − 21
If we were to try undo this multiplication through the process of trinomial factoring,
we should look to the sign of the constant term. In this example, the sign
is negative. This still means that we will need to find factor pairs that produce
a difference of (+1x) as the middle term. However, in this scenario, it is not just
the factors of 21 that are involved in producing the (+1x), but the combination
of the factors of 21 and the factors of the leading coefficient 2. The middle term
(+1x) comes from the multiplication of the (2x)(−3) and the multiplication of
(+7)(+1x):
(2x +7)(x − 3)
(2x +7)(x − 3) = 2x 2 − 6x +7x − 21
=2x 2 + x − 21
In trying to factor a trinomial like 2x 2 + x − 21, we need to take this into consideration.
For example, if we were to factor 3x 2 − 10x +8, we should first still look
to the sign of the constant term, which, in this case, is positive. That means we
want factor pairs that will add up to 10. But we have to take into consideration
the interaction of the factors of the 3 with the factors of the 8. The 3 is a prime
number, which means that we don’t have a choice - it can only be split up into
3 ∗ 1, so we can start:
Factor 3x 2 +10x +8