31
Section Lesson 6.2
Addition or Subtraction of Rational Expressions
Remember: Fractions (rationals) can only be added or subtracted if they have a common
325
denominator. For example: + =
888
To add or subtract rational expressions with the same denominator:
1. Add or Subtract the numerators.
2. Keep the denominator.
3. Factor, if possible.
4. Simplify, if possible.
A
B A B
=
C
C
C
Examples:
6 + 2x
8
1)
+
=
x+7 x+7
2)
5
7
+
=
3a 3a
3)
4y + 2 y 3
=
y2 y2
4)
3a 2
4a 7
2
=
2
a 25 a 25
1
When denominators are opposites, the denominators can be made the same polynomial
by multiplying both numerator and denominator of one rational expression by 1 .
Ex 5) Examine the following.
x7
x 1
=
2
x 16 16 x 2
x 7 x 1 1
x 2 16 16 x 2 1
( x 1)
( x 7)
=2
x 16 16 x 2
(
)
( x 7) + ( x 1)
x 2 16
x 7 + x 1 2x 8
=
=2
x 2 16
x 16
2( x 4)
2
=
=
( x + 4)( x 4) x + 4
=
When denominators are unlike, a common denominator must found be and the fractions
are rewritten by multiplying numerator and denominator by the same nonzero quantity.
To find the least common denominator (LCD):
1. Factor each denominator.
2. Find a product that contains each different factor the greatest number of times
it occurs in any denominator.
Examples:
6)
,
( x + 2) 2 x( x + 2)
7)
,
( x 3)( x + 1) ( x + 1)( x 5)
8)
,
12 x 4 + 4 x 3 54 x 4 6 x 2
2
8.5)
,
,
x 2 + 2 x x 2 ( x + 2)2 x 2 4
To add or subtract rational expressions with different denominators:
1. Find the LCD as outlined previously.
2. Multiply each numerator by the missing factor needed in order to write with the
LCD.
3. Combine numerator terms together over the common denominator.
4. Factor and simplify, if possible.
Examples:
9)
a+3 a2
+
=
a 1 a + 4
10)
3+
11)
7
9y + 2
+2
=
3y + y 4 3y 2 y 8
x+2
=
x5
2
3
12)
6 xy
x+ y
=
2
x y
x y
13)
x
5
2
=
x + 11x + 30 x + 9 x + 20
14)
2
5
a+3
+
+2
=
a+2 a2 a 4
2
2
4