2. Text Example
Find all the numbers that must be excluded from the domain of each
rational expression.
SolutionTo determine the numbers that must be excluded from each
domain, examine the denominators.
This denominator
would equal zero
if x = 2.
This denominator
would equal zero
if x = -1.
This denominator
would equal zero
if x = 1.
a.
a
x 2
b.
x
x2 1
a.
a
x 2
b.
x
x2 1
x
(x 1)(x 1)
3. Simplifying Rational Expressions
1. Factor the numerator and denominator
completely.
2. Divide both the numerator and
denominator by the common factors.
4. Example
2
4 x
• Simplify: x
8 4
Solution:
4 2
4
2
( x 2)( x
2)
x
4( 2)
4 8
x
x
x
5. Multiplying Rational Expressions
1. Factoring all numerators and
denominators completely.
2. Dividing both the numerator and
denominator by common factors.
3. Multiply the remaining factors in the
numerator and multiply the remaining
factors in the denominator.
6. x
1
x x
x x
2
3
x x
2
2 3
2
2
2
2
Example
• Multiply and simplify:
x x
2
3
x x
(2 3)
1
2
1
x x
( 1)(
1)
( 2)
(2 3)( 1)
2
2 3
x
2
2
2
2
x
x
x x
x x
x x
x x
Solution:
7. Example
x
x
2
2
3 6 3
2
• Divide and simplify: x x
2
6 24
x
x
Solution:
2
x x
x
x x
2
x x
3
6
2
2
x x
3 6
x
x
x
2 2
3
2
6 24
2
3
6 24
1
x x
3 ( 2)
6 6
1
1
1
6
2
x
3 ( 1)
6( 2)( 2)
x x
x x
x x
8. Example
3
3 1
• Add: 2
x
3 1
x x
Solution:
x
2
3
3 1
3
x
x x
3 1
2
3 1
x
9. Finding the Least Common
Denominator
1. Factor each denominator completely.
2. List the factors of the first denominator.
3. Add to the list in step 2 any factors of the
second denominator that do not appear in
the list.
4. Form the product of each different factor
from the list in step 3. This product is the
least common denominator.
10. Adding and Subtracting Rational
Expressions That Have Different
Denominators with Shared Factors
1. Find the least common denominator.
2. Write all rational expressions in terms of the least
common denominator. To do so, multiply both the
numerator and the denominator of each rational
expression by any factor(s) needed to convert the
denominator into the least common denominator.
3. Add or subtract the numerators, placing the resulting
expression over the least common denominator.
4. If necessary, simplify the resulting rational
expression.
11. Example
4
2
2
5 5
5 5
a a a
• Subtract:
Solution:
2
2
a a a
4
2
5( 1)
4
5 ( 1)
5 5
5 5
a a a
4
4
4 2
5 ( 1)
2
2
a
5( 1)
5 ( 1)
5( 1)
5 ( 1)
a
a a
a
a a
a a
a
a a a
