Least Common Denominators
Remark 1 Before we can add or subtract unlike rational expressions, we must
have a common denominator. Let’s recall the technical definition of
the least common denominator and discuss how to find it.
Definition LCD (Least Common Denominator)
The LCD for a list of rational expressions is the simplest expression that is a multiple of all denominators in the list.
Remark 2 Let’s work with usual fractions and recall how to find the LCD.
1
1
ex.
−→ Find the LCD for
and .
75
90
Since the LCD must be a multiple of both 75 and 90, it is tempting to multiply the two numbers together and say 6750 is the LCD.
While 6750 is a common denominator, it is not the LEAST common
denominator.
To find the least common denominator, we must find the smallest possible multiple of 75 and 90. To do this, we factor!
Factor denominators: Count prime factors:
• one 3
75 = 3 · 5 · 5
• two 5’s
• one 2
• two 3’s
90 = 2 · 3 · 3 · 5
• one 5
What’s needed in LCD?
• one 2
• two 3’s
• two 5’s
If we multiply the factors listed in the final column of our table we will
get a number that has all the factors of 75 and 90, but no extra factors!
Therefore, it is the LCD.
90
z
}|
{
LCD = 2 · 3 · 3 · 5 · 5 = 450
| {z }
75
Main Idea To build the LCD for a list of rational expressions:
• Multiply the prime factors of each denominator, taking
each prime factor the most times it occurs in any single
denominator.
• Build a table like the one in the previous example if it
helps.
Remark 3 The factoring method used when finding the LCD depends on the type
of denominator. When the denominators are all monomials, the LCD
is found almost exactly as it was in the previous example. However, if
the denominators have polynomials, techniques like pulling out the
GCF or reverse FOILing must be applied.
monomial
ex.
−→
Find the LCD for
7
13
and
.
8x2
6x5
Let’s factor the denominators and count our prime factors to figure out
what we need in our LCD.
Factor denominators: Count prime factors:
• three 2’s
8x2 = 2 · 2 · 2 · x · x
• two x’s
• one 2
5
6x = 2 · 3 · x · x · x · x · x • one 3
• five x’s
What’s needed in LCD?
• three 2’s
• one 3
• five x’s
Multiplying the factors in the final column, we get our LCD.
6x5
}|
{
z
LCD = 2 · 2 · 2 · 3 · x · x · x · x · x = 24x5
| {z }
| {z }
↖ 8x2 ↗
polynomial
ex.
−→
Find the LCD for
17
2x
and
.
x2 − 4
x2 + 4x + 4
To factor x2 − 4 we must apply the difference of squares formula, while
to factor x2 + 4x + 4 we reverse FOIL.
Factor denominators:
Count prime factors: What’s needed in LCD?
• one (x − 2)
• one (x − 2)
x2 − 4 = (x − 2)(x + 2)
• one (x + 2)
• two (x + 2)’s
x2 +4x+4 = (x+2)(x+2) • two (x + 2)’s
Multiplying the factors in the final column, we get our LCD.
x2 + 4x + 4
z
}|
{
LCD = (x − 2)(x + 2)(x + 2) = (x − 2)(x + 2)2
|
{z
}
x2 − 4
Example 1 Find the LCD for each list of fractions.
a
1
2
,
3y 4x
b
3
5
,
8a2b3 12a5b
c
6
3x − 1
,
x2 − 4x x2 − 16
d
7
4
,
x2 + 10x + 25 3x + 15
e
17
2
,
8−a a−8
Example 2 Rewrite each fraction to have the indicated denominator.
5t2
?
a
=
2y
16y 2
b
−7t
?
=
4t − 12 20t − 60
c
4m
?
=
m2 − 8m + 15 (m − 5)(m − 3)(m + 2)