§ 7.3
Adding and Subtracting Rational
Expressions with Common
Denominators and Least Common
Denominators
Rational Expressions
Adding and subtracting rational expressions
with common denominators when P, Q and R
are polynomials and R  0,
P Q PQ
 
R R
R
P Q P Q
 
R R
R
Martin-Gay, Beginning and Intermediate Algebra, 4ed
2
Adding Rational Expressions
Example:
Add the following rational expressions.
4p 3 3p 8 7 p  5
4 p 3 3p 8



2p 7
2p 7
2p 7 2p 7
Martin-Gay, Beginning and Intermediate Algebra, 4ed
3
Subtracting Rational Expressions
Example:
Subtract the following rational expressions.
8 y  16 8( y  2)
8y
16



 8
y2
y2 y2
y2
Martin-Gay, Beginning and Intermediate Algebra, 4ed
4
Subtracting Rational Expressions
Example:
Subtract the following rational expressions.
3y  6
3y
6

 2
 2
2
y  3 y  10
y  3 y  10 y  3 y  10
3( y  2)

( y  5)( y  2)
3
y5
Martin-Gay, Beginning and Intermediate Algebra, 4ed
5
Least Common Denominators
To add or subtract rational expressions with
unlike denominators, you have to change
them to equivalent forms that have the same
denominator (a common denominator).
This involves finding the least common
denominator of the two original rational
expressions.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
6
Least Common Denominators
Finding the Least Common Denominator (LCD)
1) Factor each denominator completely.
2) The LCD is the product of all unique factors
found in Step 1, each raised to a power equal to
the greatest number of times that the factor
appears in any one factored denominator.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
7
Least Common Denominators
Example:
Find the LCD of the following rational expressions.
1
3x
,
6 y 4 y  12
6 y  2  3y
4 y  12  4( y  3)  2 ( y  3)
2
So theLCD is 2  3 y( y  3)  12y( y  3)
2
Martin-Gay, Beginning and Intermediate Algebra, 4ed
8
Least Common Denominators
Example:
Find the LCD of the following rational expressions.
4
4x  2
, 2
2
x  4 x  3 x  10 x  21
x  4x  3  ( x  3)(x  1)
2
x  10x  21  ( x  3)(x  7)
2
So theLCD is (x  3)(x 1)(x 7)
Martin-Gay, Beginning and Intermediate Algebra, 4ed
9
Least Common Denominators
Example:
Find the LCD of the following rational expressions.
2
3x
4x
, 2
2
5x  5 x  2 x  1
5x  5  5( x 1)  5( x  1)(x 1)
2
2
x  2x  1  ( x 1)
2
2
So the LCD is 5( x + 1)( x - 1)2
Martin-Gay, Beginning and Intermediate Algebra, 4ed
10
Least Common Denominators
Example:
Find the LCD of the following rational expressions.
1
2
,
x 3 3 x
Both of the denominators are already factored.
Since each is the opposite of the other, you can
use either x – 3 or 3 – x as the LCD.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
11
Multiplying by 1
To change rational expressions into equivalent
forms, we use the principal that multiplying
by 1 (or any form of 1), will give you an
equivalent expression.
P P
P R PR
 1   
Q Q
Q R QR
Martin-Gay, Beginning and Intermediate Algebra, 4ed
12
Equivalent Expressions
Example:
Rewrite the rational expression as an equivalent
rational expression with the given denominator.
3

5
9y
72 y 9
4
3 8y
24y 4
3
 4 

5
5
9
9y 8y
9y
72y
Martin-Gay, Beginning and Intermediate Algebra, 4ed
13