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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 — 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 · 5 15 = 3 · 5 18 = 2 · 3 · 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 · 3 · 3 · 5 = 90

The LCM of 10, 15, and 18 is 90.

 

Example 1

Find the LCM of 15xy, 10x2y, and 6xy2.

Solution

Step 1 Factor each polynomial.

15xy

10x2y

6xy2

= 3 · 5 · x · y

= 2 · 5 · x · x · y

= 2 · 3 · x · y · 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 · 3 · 5 · x · x · y · y

= 30x2y2

The LCM of 15xy, 10x2y, and 6xy2 is 30x2y2.

 

Example 2

Find the LCM of x2 - 2x, x2 + x - 6, and x2 + 6x + 9

Solution

Step 1 Factor each polynomial.

 x2 - 2x

x2 + x - 6

x2 + 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  x2 - 2x, x2 + x - 6, and x2 + 6x + 9 is x(x - 2)(x + 3)(x + 3).