College Alg/Trig
Simplifying Complex Rational Expressions
Name: ________________________
Warm Up: Simplify the following. Make sure your answer is in simplest form!
3
1
+4
2
5
6
3
11
− 8 + 12
Now think about this question… How would you solve it?
𝑥 + 3 5 + 2𝑥 1
−
=
2
4
6
Is there a way you could use the method you chose to solve the second problem in the first problem? Explain.
Recall: We simplify rational expressions by _____________________________ and canceling out the
___________________________________________________.
Today, we are going to focus on simplifying complex rational expressions. Complex rational expressions (or
complex fractions) have ______________________ and ______________________ that contain one or more
rational expression. Meaning you will see fractions in multiple places!
There are two methods we can use to simplify a complex rational fraction.
Method #1: Combining the numerator to one expression and the denominator to one expression. To do this…
1. Find a ___________________________________ in both the numerator and the denominator and
perform the indicated operation in each to get a single fraction over another single fraction
2. Divide the remaining fractions using _______________, _______________, _____________________.
Method #2: Multiplying through by the _______________ __________________ ______________________ of
all the rational expressions you see!
Let’s redo the first problem from today and try method #2.
LCD? _____
3
2
5
6
+
3
1
4
11
− 8 + 12
Why is it okay to do this?
Remember: Whatever you do to the ________________________ you must also do to the
_________________________. This means you must choose the same LCD for the numerator and the
denominator. The reason for this is we really want to make sure we are only multiplying the entire fraction by
___________ because this allows us to obtain an equivalent expression that only appears different.
When looking for the least common denominator when there are variables in each denominator, it is
sometimes helpful to _______________ __________________.
Find the LCD of in each of the following.
1
1
1
−
−2
𝑥+7 𝑥
𝑥
1. 1
2.
7
2+𝑥
3.
3
4
−
𝑥−2 𝑥+2
7
𝑥2 −4
Let’s try question 1 both ways and decide which method is more efficient.
METHOD #1
METHOD #2
Now simplify questions 2 and 3 from above using the LCD method!
2.
4.
1
1
−
𝑥+7 𝑥
7
3.
𝑥−3
3
𝑥−
𝑥−2
5.
3
4
−
𝑥−2 𝑥+2
7
𝑥2 −4
1
𝑥+1
1
1
+
𝑥2 −2𝑥−3 𝑥−3