Rationalexpressionsandequations 100706140157-phpapp01

Copyright © 2007 Pearson Education, Inc. Slide R-1

Rational Expressions and Equations

Chapter 9

pg 470


Review of Rational Expressions

• A rational expression is an expression that is the quotient of two polynomials.

Examples include

2

2

6 ( 6)( 4) 2 7 4, ,

2 ( 2)( 4) 5 20

x x x p p

x x x p p


Domain of a Rational Expression

• The domain of a rational expression is the set of real numbers for which the expression is defined.

• The domain consists of all real numbers except those that make the denominator 0.


Domain of a Rational Expression

For example, to find the domain of

solve as follows,

or

or

The domain is

( 6)( 4)

( 2)( 4)

x x

x x

( 2)( 4) 0x x

2 0x 4 0x

2x 4x

{ | 2, 4} .x x


Lowest Terms of a Rational Expression

Fundamental Principle of Fractions

( 0 , 0)ac a

b cbc b


Writing Rational Expressions in Lowest Terms

Example Write each rational expression in lowest terms.

(a) (b)

Solution

(a)

by the fundamental principle, provided p is not 0 or –4.

2

2

2 7 4

5 20

p p

p p

2

2

2 7 4 (2 1) 2 1( 4)

5 20 5 5( 4)

p p p p

p p p p

p

p

2

6 3

4

k

k


Writing Rational Expressions in Lowest Terms

Solution

(b)

by the fundamental principle.

2

6 3 3 3

4 ( 2) ( 2

(2 ) (2 )( 1)

( 2) ( 2)( 1)

(

)

3 3

( 2)

2 )( 1)

2) ( )(2

k k

k k

k

k

k

k k k

k k


Multiplying and Dividing Rational Expressions

Multiplying and Dividing Fractions

For fractions and

and

( 0 , 0),c

b dd

a

b

a c ac

b d bd , if 0.

a c a d c

b d b c d



Example Multiply or divide as indicated.

(a) (b)

Solution

(a)

2

3 2 4 3

3 11 4 9 36

24 8 24 36

p p p

p p p p

2

5

2 27

9 8

y

y

2

5 3

3

2

2

2

5

22

2

27 2 27 3

9 8 9 8

3

4

9 4

9 y

y

y y

y y y

y



Solution (b)

2

3 2 4 3 2 3

3

2

3

2

3 11 4 9 36 ( 4)(3 1) 9( 4)

24 8 24 36 8 (3 1) 12 (2 3)

(12 )(2 3)

8 (9)

12 (2 3)

9 8

(2 3)

( 4)

(

(3 1)

(3 1 ))

6

4

p p p p p p

p p p p p p p p

pp

p

p

p

p

p

p p

p

p p


Complex Fractions

• Complex fractions are those fractions whose numerator & denominator both contain fractions.


Now you try!

• Pg 476- 477

• #’s 14-22 evens

• #’s 26-34 evens

• #’s 36, 37, 38


Adding and Subtracting Rational Expressions

Adding and Subtracting Fractions

For fractions and

and

( 0 , 0),c

b dd

a

b

a c ad bc

b d bd

.

a c ad bc

b d bd

•Addition and subtraction are typically performed using the least common denominator.



Finding the Least Common Denominator (LCD)

1. Write each denominator as a product of prime factors.

2. Form a product of all the different prime factors. Each factor should have as exponent the greatest exponent that appears on that factor.


Adding/Subtracting

• when we talk about CDs, we mean denominators that contain the same factors.

• To find our CD, we will first factor the ones we have.

• Then we will multiply each denominator by the factors it is missing to create a CD.

• Remember, we must also multiply the numerator by that same factor.



Example Add or subtract, as indicated.

(a) (b)

Solution

(a) Step 1: Find the LCD

2 2

2 3

2 4 2

y y

y y y y

2

5 1

9 6x x



Solution (a) The LCD is

Then1 2 2 22 3 18 .x x

2 2

2 2

2

5 1 5 1

9 6 9 610 3

18 181

2

.

2

0

3

3

3

18

x

xx x x xx

x xx

x



Solution (b)

2 2 2

2

2 2 2 2

2 2 2

2

2

2 3 2 3

2 4 2 ( 1) 2( 1)

( 2) 3

( 1) 2( 1)

2( 2) 3 2 2 4 3

2 ( 1) 2 ( 1) 2

2( 1

( 1)

2 4

)

2(

2 (

)

1

1

)

y y y y

y y y y y y y

y y

y y y

y y y y y y

y y y y y y

y y

y

y

y

y

y y


Now you try!

• Pg 482- 483 #’s 26-36 evens


Complex Fractions

• A complex fraction is any quotient of two rational expressions.


Simplifying Complex Fractions

Example Simplify

Solution

Multiply both numerator and denominator by the LCD of all the fractions a(a + 1).

11

1 11

aa a

a a


Simplifying Complex Fractions

Solution

2 2

( 1) ( 1)11 1

11 11 1 1 11 1

1 1

( 1)

( 11

( 1) 1

( 1) 2

) 1

1

( )( 1)

a a a a aaa aa aa a a a

a a a aa a

a a a a

a a

a

a a

a

a aa a


Now you try!

• Pg 483 #’s 40

Rationalexpressionsandequations 100706140157-phpapp01

Technology

x x x p p x x x p p

x x x x x x x x

slide r

pearson education

b solution

rational expressions

d c b d b c d

c b d d