P6Rational Expressions

Warm-upSimplify the following:

Rational expression

•  

• Activity: Graph the expression above in your calculator. And fill in the table:

• Do you notice anything unusual? Discuss possible reasons for your result with your neighbor.

X Y

0

1

2

3

4

𝑥+5

𝑥2−25

Example 1: Excluding Numbers from the Domain

•  

Example 2: Simplifying Rational Expressions

•  

Example 3: Multiplying Rational Expressions

•  

Example 4: Dividing Rational Expressions

•  

Text Example

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.

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

• Simplify:

84

42

x

x

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

• Divide and simplify:

2

3

246

63 2

2

2

x

xx

x

xx

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• Add:

13

3

13

2

xx

x

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 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.

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