5.3 Least Common Denominator

1
5.3 Least Common Denominator
In order to add or subtract rational expressions
we need to find the least common denominator
(LCD) for all the expressions.
Let us consider the problem
To find the LCD , we factor each denominator.
6 3 2
4 2 2
15 3 5
2
6 3 2
4 2 2
15 3 5
To obtain the LCD, we pick all the factors
without repeating
We start by picking 3 2.
In the second number out of 2 2 we pick only
one 2.
From the third number out of 3 5 we pick only 5.
The LCD is 3 2 2 5
3
Find the least common denominator for
The first denominator is 16 x2 y
The second denominator is 3 x3
When there are no common factors, the LCD of two
numbers is their product.
The lowest common multiple of 16 and 3 is 16 3
48
The lowest common multiple of x2 and x3 is x3
The lowest common multiple of y and 1 is y
The LCD is 48x3y
4
Find the least common denominator for
The LCD contains all the factors of the
denominator without repeating.
The first denominator is (r 7) (r 3)
We start by picking both these factors.
The second denominator is (r 7)
Since (r 7) has already been picked, we dont
need this.
The LCD is (r 7) (r 3)
5
Find the least common denominator for
First we factor each denominator.
The first denominator is 8k
The second denominator is 12k 24
12(k 2)
First we obtain the LCD for 8 and 12
8 4 2
12 4 3
The LCD of 8 and 12 is 4 2 3 24
The LCD of k and (k 2) is k(k 2)
The LCD for the whole expression is 24k(k 2)
6
Find the least common denominator for
First we factor each denominator
The first denominator is 7x2 21x
7x(x 3)
The second denominator is 5x2 15x
5x(x 3)
The LCD of 7 and 5 is 7 5 35
The LCD of x and x is x
The LCD of (x 3) and (x 3) is (x 3)
The LCD for the whole expression is 35x(x 3)
7
Find the least common denominator for
Factor each denominator
2x2 7x 3 (2x 1) (x 3)
We pick both these factors.
4x2 4x 3 (2x 3) (2x 1)
We pick (2x 3)
2x2 3x 9 (2x 3) (x 3)
We have picked both these already.
The LCD is (2x 1) (2x 3) (x 3)
8
Find the least common denominator for
Factor each denominator
z2 4z 12 (z 6) (z 2)
We pick both these factors.
z2 z 30 (z 6) (z 5)
We pick (z 5)
z2 2z 24 (z 6) (z 4)
We pick (z 4)
The LCD is (z 6) (z 2) (z 5)(z 4)