MTH 092 — Summer I 2008, NSF Special Session
Essex County College — Division of Mathematics
Worksheet #201 — April 27, 2008
1
Chapter 6, Least Common Multiples, Finding and Using
1.1
Finding an LCM
The least common multiple (LCM) of two or more polynomials is the polynomial of least
degree that contains all the factors of each polynomial. To find the LCM, first factor each
polynomial completely. Then, the LCM is the product of each factor the greatest number of
times it occurs in any one factorization.
Let’s first take a simple numerical example. Find the LCM of 50 and 60.
First factor.
50 = 2 · 5 · 5
60 = 2 · 2 · 3 · 5
The LCM is:
22 · 3 · 52 = 300.
Now let’s take a more complicated example. Find the LCM of x2 − 6x + 9 and x2 − 2x − 3.
First factor.
x2 − 6x + 9 = (x − 3) · (x − 3)
x2 − 2x − 3 = (x − 3) · (x + 1)
The LCM is:
(x − 3)2 · (x + 1) .
No need to multiple it out.
1.2
Using an LCM
The initial purpose2 of finding an LCM is so that we can rewrite fractions in terms of least
common denominator (LCD). The basic idea here is that
a
a
a c
ac
= ·1= · = .
b
b
b c
bc
So if you have a fraction
9
5
and you want to rewrite the fraction with another denominator, say 25, just do this:
9
9
9 5
9·5
45
= ·1= · =
= .
5
5
5 5
5·5
25
1
2
This document was prepared by Ron Bannon using LATEX 2ε .
Will also use it to simplify and solve rational equations.
1
Now let’s take a more complicated example. So if you have a fraction
15x
13y
and you want to rewrite the fraction with another denominator, say 26xy 2 , just do this:
15x
15x
15x 2xy
15x · 2xy
30x2 y
.
=
·1=
·
=
=
13y
13y
13y 2xy
13y · 2xy
26xy 2
1.3
Examples
1. Find the LCM of the polynomials.
6x2 y,
18xy 2
2. Find the LCM of the polynomials.
6x2 ,
4x + 12
3. Find the LCM of the polynomials.
8x2 (x − 1)2 ,
2
10x3 (x − 1)
(2x − 1)2 (x − 5)
4. Find the LCM of the polynomials.
(2x − 1) (3 − 5x) ,
5. Find the LCM of the polynomials.
x2 − 2x − 24,
x2 − 36
6. Find the LCM of the polynomials.
2x2 − 7x + 3,
2x2 + x − 1
7. Find the LCM of the polynomials.
x2 + 3x − 18,
x − 3,
8. Find the LCD.
4
,
x
3
x2
3
x−2
9. Rewrite the fractions in terms of the LCD.
4
,
x
3
x2
10. Rewrite the fractions in terms of the LCD.
a2
,
x (x + 7)
a
(x + 7)2
11. Rewrite the fractions in terms of the LCD.
3
,
x (x − 5)
2
(x − 5)2
4
12. Rewrite the fractions in terms of the LCD.
13. Rewrite the fractions in terms of the LCD.
14. Rewrite the fractions in terms of the LCD.
5
x2
7
,
+x−2
3x
,
x−4
x2
x
x+2
5
x2 − 16
x−1
,
+ 2x − 15
x2
x
+ 6x + 5