§7.3 ADDING AND SUBTRACTING RATIONAL EXPRESSION WITH COMMON DENOMINATORS
AND LEAST COMMON DENOMINATOR (LCD)
In §7.2 we learned:
 To multiply and divide rational expressions, we use the same methods that we used to multiply and divide
fractions.
 To multiply rational expressions, first factor the numerators and denominators completely. Then multiply the
numerators (straight across), multiply the denominators (straight across), and then simplify (divide out common
factors.
 To divide rational expressions, multiply the first by the reciprocal of the second. Reciprocal means the same for
rational expressions as it does for fractions – flip it over. By “flip it over” I mean to switch the numerator and the
denominator.
ADDING AND SUBTRACTING RATIONAL EXPRESSIONS WITH COMMON DENOMINATORS
Just like multiplication/division of rational expressions is very similar to multiplication/division of fractions, we
add/subtract rational expressions just like we add/subtract fractions.
Remember, to add and subtract two (or more) fractions, they must have common denominators. If they have a common
denominator, then we simply add/subtract the numerators and then place the sum/difference over that common
denominator.
Ex.
3 2 3 2 5
 

7 7
7
7
Don’t forget to simplify (if possible) after we add/subtract the numerators.
Ex.
3
2 3 2 5 1




10 10
10
10 2
If you have a common denominator, don’t simplify before you add/subtract the numerators. If you do, you’ll no longer
have a common denominator.
Ex.
3 2
3 1
  
10 10 10 5
Although this is true, we lost our common denominator and will have to get it back again
before we can perform the operation.
To add/subtract rational expressions that have a common denominator add/subtract the numerators, place the
sum/difference over the common denominator, and then simplify (if possible).
Ex.
9
3
93
12



x2 x2 x2 x2
Practice 1, page 451
Add:
7a a

4b 4b
Since 12 and (x+2) have no common factors, we can’t simplify our answer.
Practice 2, page 451
Subtract:
3x
2

3x  2 3x  2
Practice 3, page 451
Subtract:
4 x 2  15 x 8 x  15

x3
x3
FINDING THE LCD (LEAST COMMON DENOMINATOR) OF RATIONAL EXPRESSIONS
In the above examples, we were adding rational expressions that already had a common denominator. Just as with
fractions, if we want to add rational expressions with unlike denominators, we first must find the LCD.
Ex. Suppose we want to calculate
5
9

. We must first find the LCD. We then rewrite the fractions as equivalent
12 42
fractions, each over the LCD. Finally, we add the numerators, place the sum over the LCD and then simplify.
To find the LCD of fractions, we look at the prime factorizations of the denominators. (12 = 223 and 42 = 237)
The LCD is the smallest number which is a multiple of both 12 and 42; it is the LCM (Least Common Multiple) of the
denominators. . Another way to say this is the LCD is the smallest number that is divisible by both 12 and 21 with no
remainder. This means that the LCD must contain all factors of 12 and all factors of 42. So our LCD is 2237 = 84.
Our LCD must contain every factor that appears in each factorization, and we use it the greatest number of times it
appears in any one factorization.
We now rewrite our equation with equivalent fractions that have 84 as their denominators.
5
9 5 7 9 4 35 36 35  36 71

 



=  
12 42 12 7 42 4 84 84
84
84


Multiplying by 1 gives an equivalent fraction.
Using 7/7 and 4/4 will give us fractions over 84.
HOW TO FIND THE LCD OF RATIONAL EXPRESSIONS
1. Completely factor the denominator of each expression
2. Form a product of all unique factors found in Step 1 and raise each one to a power equal to the greatest
number of times that the factor appears in any one factored denominator.
(If Step 2 is difficult to understand in a sentence, it will probably make more sense after we do some examples)
Practice 4, page 452, part b
Find the LCD of the given pair of rational expressions.
4 11
,
9 y 15 y 3
9y = 33y = 32y
STEP 1: Factor the denominators
15y3 = 35yyy = 35 y3
STEP 2: The unique factors from step are: 3, 5, y
The greatest number of times that the factor 3 appears in any one factorization is 2 so 32 is a factor of our LCD.
The greatest number of times that the factor 5 appears in any one factorization is 1 so 51 = 5 is a factor of our LCD.
The greatest number of times that the factor y appears in any one factorization is 3 so y3 is a factor of our LCD.
Therefore, the LCD = 32 5 y3 = 45y3
Practice 5, page 453. Find the LCD of
a.
16 3 y 3
,
y 5 y 4
b.
8 5
,
a a2
Practice 6, page 453. Find the LCD of
2x3
5x
,
2
(2 x  1) 6 x  3
Practice 7, page 453. Find the LCD of
x5
x8
, 2
x  5 x  4 x  16
Practice 8, page 454 Find the LCD of
2
5
4
,
3 x x 3
Here are some things to think about. We’ll talk more about them in the next section when we use the LCD to
add/subtract rational expressions. (These are questions from the exercise set in §7.4)
#85, page 463 Explain (in words) when the LCD of the rational expressions in a sum is the product of the denominators.
#86, page 463 Explain (in words) when the LCD is the same as one of the denominators of a rational expression to be
added or subtracted.
If you’re not sure of the answers to these questions, look at the examples we just did and do some homework problems.
WRITING EQUIVALENT RATIONAL EXPRESSIONS
In order to use the LCD, we have to be able to use it. This means we have to be able to write rational expressions as an
equivalent rational expression with a given denominator. To do this we multiply by a form of 1.
Practice 9, page 454 Write each rational expression as an equivalent rational expression with the given denominator.
a.
3x

5 y 35 xy 2
b.
9x

4 x  7 8 x  14
Practice 10, page 455 Write the rational expression as an equivalent rational expression with the given denominator.
3

x  2 x  15 ( x  2)( x  3)( x  5)
2
CHALLENGE QUESTIONS: If the examples we did in class were easy for you, try #83 and #84 on page 457. The answer
to 83 is in the back of your book. The approach to finding the answer is more interesting than the actual number. See
if you can explain how to find the answer and why this is the method to use. I WON’T put questions like this on the
test or on the homework and these are NOT required problems.