Finding the Least Common Multiple (LCM)
of a Set of Polynomials
Adding and subtracting rational expressions with different denominators is
like adding and subtracting fractions with different denominators.
We begin by finding the least common denominator (LCD) of a set of
rational expressions.
The LCD of two or more rational expressions is the least common
multiple (LCM) of their denominators.
We can find the LCM of a set of polynomials in much the same manner
that we found the LCM of a set of whole numbers.
Procedure â€”
To Find the Least Common Multiple (LCM) of a Set of Polynomials
Step 1 Factor each polynomial.
Step 2 For each factor, list it the greatest number of times it appears
in any factorization.
Step 3 Find the product of the factors in the list.
We usually leave the LCM in factored form.
Note:
To find the LCM of a set of numbers, say
10, 15, and 18, follow these steps:
Step 1 Write the prime factorization of
each number.
10 = 2 Â· 5
15 = 3 Â· 5
18 = 2 Â· 3 Â· 3
Step 2 List each prime factor the greatest
number of times it appears in any
factorization:
2, 3, 3, 5
Step 3 Multiply the prime factors in the
list:
2 Â· 3 Â· 3
Â· 5 = 90
The LCM of 10, 15, and 18 is 90.
Example 1
Find the LCM of 15xy, 10x^{2}y, and 6xy^{2}.
Solution
Step 1 Factor each polynomial.

15xy
10x^{2}y
6xy^{2} 
= 3 Â· 5 Â· x Â· y
= 2 Â· 5 Â· x Â· x Â· y
= 2 Â· 3 Â· x Â· y Â· y 
Step 2 For each factor, list it the
greatest number of times
it appears in any factorization.


2, 3, 5, x, x, y, y 
Step 3 Find the product of the factors
in the list. 
LCM 
= 2 Â· 3
Â· 5 Â· x Â· x Â· y Â· y
= 30x^{2}y^{2} 
The LCM of 15xy, 10x^{2}y, and 6xy^{2}
is 30x^{2}y^{2}.
Example 2
Find the LCM of x^{2}  2x, x^{2} + x  6, and x^{2}
+ 6x + 9
Solution
Step 1 Factor each polynomial. 
x^{2}  2x
x^{2} + x  6
x^{2} + 6x + 9 
= x(x  2) = (x + 3)(x  2)
= (x + 3)(x + 3) 
Step 2 For each factor, list it
the greatest number of
times it appears in any
factorization.


x, (x  2), (x + 3), (x + 3) 
Step 3 Find the product of the
factors in the list. 
LCM 
= x(x  2)(x + 3)(x + 3) 
The LCM of x^{2}  2x, x^{2} + x  6, and x^{2}
+ 6x + 9 is x(x  2)(x + 3)(x + 3).