Chapter 7 Rational Functions - College of the Redwoods

684 Chapter 7 Rational Functions
The next step is to create equivalent fractions using the LCD as the denominator.
So, in the case of 5/12,
In the case of 5/18,
5
12 = 5 12 · 1 = 5
12 · 3
3 = 15
36 .
5
18 = 5 18 · 1 = 5
18 · 2
2 = 10
36 .
If we replace the fractions in equation (8) with their equivalent fractions, we can
then add the numerators and divide by the common denominator, as in
5
12 + 5 18 = 15
36 + 10 15 + 10
= = 25
36 36 36 .
Let’s examine a method of organizing the work that is more compact. Consider the
following arrangement, where we’ve used color to highlight the form of 1 required to
convert the fractions to equivalent fractions with a common denominator of 36.
5
12 + 5 18 = 5 12 · 3
3 + 5 18 · 2
2
= 15
36 + 10
36
= 25
36
Let’s look at a more complicated example.
◮ Example 9.
State all restrictions.
Simplify the expression
x + 3
x + 2 − x + 2
x + 3 . (10)
The denominators are already factored. If we take each factor that appears to the
highest exponential power that appears, our least common denominator is (x+2)(x+3).
Our first task is to make equivalent fractions having this common denominator.
x + 3
x + 2 − x + 2
x + 3 = x + 3
x + 2 · x + 3
x + 3 − x + 2
x + 3 · x + 2
x + 2
= x2 + 6x + 9
(x + 2)(x + 3) − x2 + 4x + 4
(x + 2)(x + 3)
Now, subtract the numerators and divide by the common denominator.
Version: Fall 2007