College Algebra & Trigonometry, 2018a

56 CHAPTER 1. ALGEBRA REVIEW
the 4 is positive and the x is negative, whereas in the second binomial, the 4 is
negative and the x is positive. So, we know that 4 − x ≠1. However, if we factor
x − 4
a (−1) out of the numerator, we will see an interesting phenomenon:
4 − x
x − 4 = −1(−4+x)
x − 4
=
−1(x − 4)
x − 4
= −1 ∗ x − 4
x − 4
= −1 ∗ 1=−1
Therefore, although 4 − x
4 − x
≠1, we can say that = −1. This will allow us to
x − 4 x − 4
cancel (4 − x) and (x − 4) and replace them with (−1).
16 − x 2
x 2 + x − 20
=
(4 + x)(4 − x)
(x +5)(x − 4)
= (4 + x) ✘ ✘✘ ✘
(4 − x)
(x +5) ✘ (x✘✘✘
− 4)(−1)
In the final answer, the (−1) can be placed in the denominator or the numerator,
but not both. It can also be placed in front of the fraction.
4+x
−1(x +5)
=
−1(4 + x)
x +5
= − 4+x
x +5