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).
|