Rational Expressions

Find all the numbers that must be excluded from the domain of each rational expression.

This denominator would equal zero if x = 2.

This denominator would equal zero if x = -1.

This denominator would equal zero if x = 1.

SolutionTo determine the numbers that must be excluded from each domain, examine the denominators.

Text Example

a.a

x 2b.

x

x2 1

a.a

x 2b.

x

x2 1

x

(x 1)(x 1)

Simplifying Rational Expressions

1. Factor the numerator and denominator completely.

2. Divide both the numerator and denominator by the common factors.

Example

84

42

x

x• Simplify:

Solution:

4

2

)2(4

)2)(2(

84

42

x

x

xx

x

x

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.

xx

x

xx

xx

2

1

32

322

2

2

2

Example

• Multiply and simplify:

2

1

)2(

)1)(1(

)1)(32(

)32(2

1

32

322

2

2

2

x

x

xx

xx

xx

xxxx

x

xx

xxSolution:

Example

2

3

246

63 2

2

2

x

xx

x

xx• Divide and simplify:

Solution:

xx

x

x

xx

x

xx

x

xx

22

2

2

2

2

3

2

246

63

2

3

246

63

66

1

1

1

6

1

)1(3

2

)2)(2(6

)2(3

xx

xx

x

xx

xx

Example

13

3

13

2

xx

x• Add:

Solution:

13

32

13

3

13

2

x

x

xx

x

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.

Adding and Subtracting Rational Expressions That Have Different

Denominators with Shared Factors1. 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.

55

2

55

42

aaa

Example

• Subtract:

Solution:

)1(5

2

)1(5

455

2

55

42

aaa

aaa

)1(5

24

)1(5

2

)1(5

4

)1(5

2

)1(5

4

aa

a

aa

a

aa

a

a

aaa

Rational Expressions