Chapter 8: Rational Expressions
8.3 Least Common Denominators
Finding the least common denominator, or LCD, is very important to working with rational expressions. The process used depends on finding what is common to each rational expression and identifying what is not common. These common and not common factors are then combined to form the LCD.
Example 8.3.1
Find the LCD of the numbers 12, 8, and 6.
First, break these three numbers into primes:
Then write out the primes for the first number, 12, and set the LCD to
Notice the factorization of 8 includes
Now the LCD =
Checking
The LCD =
This process can be duplicated with variables.
Example 8.3.2
Find the LCD of
First, break both terms into primes:
Then write out the primes for the first term,
The LCD for
This yields LCD =
This process can also be duplicated with polynomials.
Example 8.3.3
Find the LCD of
First, we factor both of these polynomials, much like finding the primes of the above terms:
The LCD is constructed as we did before, except this time, we write out the factored terms from the first polynomial, so the LCD =
Notice that
The LCD =
Questions
For Questions 1 to 10, find each Least Common Denominator.
For Questions 11 to 20, find the LCD of each fraction and place each expression over the same common denominator.