1) To add or subtract rational expressions with the same or different denominators, we first find the least common denominator (LCD), which is the least common multiple of the denominators. 2) We then rewrite the rational expressions with the LCD as the common denominator before adding or subtracting the numerators. 3) Once the rational expressions have been rewritten with the same denominator, we can perform the addition or subtraction operation as we would with numerical fractions.

2. Adding and subtracting Rational Expressions
with the Same Denominator
Let a, b, and c be polynomials where
Algebra:
a b a+b
+ =
c c
c
c ¹ 0.
a b a-b
- =
c c
c
2 3 2+3 5
Examples:
=
+ =
x
x x
x
5
2 5- 2 3
=
=
4x 4x
4x 4x

3. Example 1
Add and subtract with the same denominator
5
7
12
a.
+
=
3x
3x
3x
Add numerators.
3•4
=
3•x
Factor and divide out common
factor.
4
=
x
Simplify.
x +5
3x – ( x + 5 )
–
b.
=
x – 1
x – 1
x –1
3x
2x – 5
=
x – 1
Subtract numerators.
Simplify.

4. Least Common Denominator
The least common denominator (LCD) of two or more
rational expressions is the same as the least common
multiple of the denominators of the rational
expressions.

5. Example 2
Find the LCD of rational expressions
Find the LCD of the rational expressions.
1 r +3
,
a.
4r 10r 2
5
3x + 4
,
b.
( x – 3 )2 x2 – x – 6
3
, c + 8
c.
c – 2 2c + 7

6. Example 2
Find the LCD of rational expressions
SOLUTION
a. Find the least common multiple ( LCM ) of 4r and 10r 2.
4r = 22 • r
10r2 = 2 • 5 • r 2
LCM = 22 • 5 • r 2 = 20r 2
ANSWER
1
r +3
The LCD of
and
is 20r2.
10r2
4r

7. Example 2
Find the LCD of rational expressions
b. Find the least common multiple ( LCM ) of ( x – 3 )2
and x 2 – x – 6.
( x – 3 )2 = ( x – 3 )2
x2 – x – 6 = ( x – 3 ) • ( x + 2 )
LCM = ( x – 3 )2 ( x + 2 )
ANSWER
3x + 4
5
The LCD of
and
is ( x – 3 )2 ( x + 2).
( x – 3 )2
x2 – x – 6

8. Example 2
Find the LCD of rational expressions
c. Find the least common multiple of c – 2 and 2c + 7.
Because c – 2 and 2c + 7 cannot be factored, they
don’t have any factors in common. The least
common multiple is their product, ( c – 2 ) ( 2c + 7 ).
ANSWER
3
c + 8
The LCD of
and
is (c – 2)( 2c + 7).
c– 2
2c + 7

9. Different Denominators
To add or subtract rational expressions that have
different denominators, use the LCD to write equivalent
rational expressions that have the same denominator
(just as you would for numerical fractions). In the
example below, 12x 2 is the LCD.

10. Example 3
Add expressions with different denominators
Find the sum
9
5
.
+
8x 2
12x 3
9 • 3x
5 • 2
9
5
=
+
+
2
3
2 • 3x
8x
12x
8x
12x 3 • 2
Rewrite fractions
using LCD, 24x 3.
27x
10
=
+
3
24x
24x 3
Simplify numerators
and denominators.
27x + 10
=
24x 3
Add fractions.

11. Example 4
Subtract expressions with different denominators
Find the difference
10
7x
.
–
3x
x + 2
10
7x
10 ( x + 2 )
7x ( 3x )
Rewrite fractions
–
–
=
( x + 2 ) ( 3x ) using LCD, 3x ( x + 2 ).
3x ( x + 2 )
3x
x + 2
10 ( x + 2 ) – 7x ( 3x )
=
3x ( x + 2 )
=
–21x 2 + 10x + 20
3x ( x + 2 )
Subtract fractions.
Simplify numerator.

12. Example 5
Subtract expressions with different denominators
x + 4
x – 1
–
Find the difference 2
.
2 + 2x – 8
x + 3x – 10
x
x + 4
x – 1
–
2 + 3x – 10
x
x2 + 2x – 8
x + 4
x – 1
–
Factor denominators.
=
( x – 2 ) ( x + 5 ) ( x + 4 )( x – 2 )
( x + 4 )( x + 4 )
( x – 1 )( x + 5 )
–
=
( x + 4 )( x – 2 )( x + 5 )
( x – 2 )( x + 5 )( x + 4 )
Rewrite fractions using
LCD, ( x – 2 ) ( x + 5 ) ( x + 4 ).