College Algebra & Trigonometry, 2018a

1.2. FACTORING 25
As we see, factoring x 2 − 36 means that the factors of the perfect square 36 = 6 ∗ 6
will cancel each other out leaving 0x in the middle. If there is a perfect square as
the leading coefficient, then this number should be square rooted as well:
16x 2 − 25 = (4x + 5)(4x − 5)
In the example above, the +20x and −20x as the middle terms cancel each other
out leaving just 16x 2 − 25.
These three types of factoring can also be combined with each other as we see in
the following examples.
Example
Factor 2x 2 − 50
This is not a trinomial because it doesn’t have three terms. It is also not a difference
of squares because 2 and 50 are not perfect squares. However, there is a
common factor of 2 which we can factor out:
2x 2 − 50 = 2(x 2 − 25)
The expression inside the parentheses is a difference of squares and should be
factored:
Example
Factor 24 − 2x − x 2
2x 2 − 50 = 2(x 2 − 25) = 2(x +5)(x − 5)
Here the sign of the x 2 term is negative. For this problem we can factor out
a −1 and proceed as we did with the previous problems in which the leading
coefficient was positive or we can factor it as it is:
24 − 2x − x 2 = −(x 2 +2x − 24) = −(x +6)(x − 4)