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Chapter 7 - Rational Expressions and Functions

• 7-1 Rational Expressions and Functions;

Multiplying and Dividing

• 7-2 Adding and Subtracting Rational Expressions

• 7-4 Equations with Rational Expressions and Graphs

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7-1 Rational Expressions and Functions;Multiplying and Dividing

Defining rational expressions.

A Rational Expression is the quotient of two polynomials with the denominator not equal to zero.

Note: Rational Expressions are the elements of the set:

• Example 1: Rational Expressions

, , 0PP Q polynomials with Q

Q

25

2

m+4 8x 2 5, , , x

m-4 4x 5

x x

y x

3

7-1 Rational Expressions and Functions;Multiplying and Dividing

Defining rational functions and describing their domain.

A function that is defined by a rational expression is called a Rational Function and has the form:

Note: The domain of a rational function contains all real numbers except those that make Q(x) = 0.

• Example 2: Find all numbers that are in the domain of each rational function

( )( ) ( ) 0

( )

P xf x with Q x

Q x

Solution: 5 0

5 All real numbers e

2(

xcept 5

)5

f xx

x

x x

4

7-1 Rational Expressions and Functions;Multiplying and Dividing

Defining rational functions and describing their domain.

A function that is defined by a rational expression is called a Rational Function

• Example 3: Find all numbers that are in the domain of each rational function

22

Solution: 6 0

( 3)( 2) 0

3 0 2 0

3 2

A

6(

ll r

)6

ea

x x

x x

x x

x x

xf x

x x

l numbers except 3 2x and x

5

7-1 Rational Expressions and Functions;Multiplying and Dividing

Defining rational functions and describing their domain.

Example 4: Find all numbers that are in the domain of each rational function

Solution:

All r

6( )

5eal numbers

0

5x

f x

22

2 Note: There are no real numbers whose square =

Solution: 6 0

6

All rea

-

2( )

l

6

numb

6

ers

f x

x

xx

6

7-1 Rational Expressions and Functions;Multiplying and Dividing

Writing rational expressions in lowest terms.

Fundamental Property of of Rational Numbers:

If is a rational number and c is any nonzero number, then

This is equivalent to multiplying by 1.

Note: A rational expression is a quotient of two polynomials. Since the value of a polynomial is a real number for every value of the variable for which it is defined, any statement that applies to rational numbers also applies to rational expressions.

a

b a ac

b bc

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7-1 Rational Expressions and Functions; Multiplying and Dividing

Writing a rational expression in lowest terms.

Step 1: Factor both the numerator and denominator to find their greatest common factor (GCF).

Step 2: Apply the fundamental property.

• Example 4: Write each rational expression in lowest terms.2

2

2

2

3 2 22

8 x x) = 1

8 6 6

( 1) 3 3) =

8 =

48 6

2 3 ( 3) =

( 2)3 2

2 =

4

1 ( 1) ( 1) =

1( 1) 2 2

) Already in lowest terms

( 1)) = 1 1

( 11 ) 1

xA

x

x x xB

x x x

C

xD x

x x

x

x x x

x

xx

x x

y

y

x x x x x

x

8

7-1 Rational Expressions and Functions; Multiplying and Dividing

Writing a rational expression in lowest terms.

Example 5: Write each rational expression in lowest terms.

Note: In general, if the numerator and the denominator are opposites, then the expression equals -1.

2 2

2 ( 2) =

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

( 2) 1) =

( 2) 2

xA

x x

x xx xx

2 2= = -1

2 ( 1)( 2)

x x

x x

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7-1 Rational Expressions and Functions;Multiplying and Dividing

Multiplying rational expressions. Step 1: Factor all numerators and denominators as completely as possible

Step 2: Apply the fundamental property

Step 3: Multiply remaining factors in the numerator and denominator, leaving the denominator in factored form.

Step 4: Check to make sure the product is in lowest terms.

Example 6: Multiply2 2

2 2

( 2) ( 2)( 2) Solution:

( 2)( 2) ( 1)

( 2)( 2) ( 2)( 2)

2 4 4

( 2

4

)

( 1( 2) ( 1) ( 2) ( 1) )

x x x x

x x x x

x x x x x x

x

x x x x

x

xx x x

x

x

x

x

x

10

7-1 Rational Expressions and Functions;Multiplying and Dividing

Multiplying rational expressions. Example 7: Multiply

2 ( 4)( 4) 1 Solution:

( 2) ( 4)

(

16 1

( 4)( 4) 4)

( 2)

4

( 4)

2

2)(

xx

x x

x

x x

x x

x

x

x x

11

7-1 Rational Expressions and Functions;Multiplying and Dividing

Finding reciprocals for rational expressions.

To find the reciprocal of a non-zero rational expression, invert the rational expression.

• Example 8: Find all reciprocal

Rational Expression Reciprocal

2

2

5

2

5

9

20

9

unde fined 4

x

m m

m

x

m

12

7-1 Rational Expressions and Functions;Multiplying and Dividing

Dividing rational expressions. To divide rational expressions, multiply the first by the reciprocal of the second.

Example 9: Divide5 2 5

Solution: 6 15 6

5 2 5 5(5 2)

6 3(5 2) 18(5 2)

5 2 15 6

5

1

6

8

5

x

xx x

x x

x x

2

2

2

2 ( 2) 3 6 Solution:

5 4( 2) 3( 2) 3 ( 2)( 2)

5 (5 )

2 4

5 3 63

5( 1)( 2)( 2)( 4)

x x x

x

x x x

x xx x x x x x

x x

x

xx xx

x

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7-2 Adding and Subtracting Rational Expressions

Adding or subtracting rational expressions with the same common denominator.

Step 1: If the denominators are the same, add or subtract the numerators and place the result over the common denominator.

If the denominators are different, find the least common denominator. Write the rational expressions with the least common denominator and add or subtract numerators. Place the result over the common denominator.

Step 2: Simplify by writing all answers in lowest terms

• Example 1: Add or subtract as indicated

2 22

2 1=

2( 1)

( 2)( 1

2 1 -

)

2 21

2

x x

x

x x

x x x x xx

x x x

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7-2 Adding and Subtracting Rational Expressions

Adding or subtracting rational expressions with different denominators. Step 1: If the denominators are different, find the least common denominator. Write the rational expressions with the least common denominator and add or subtract numerators. Place the result over the common denominator.Step 2: Simplify by writing all answers in lowest terms

• Finding the Least Common Denominator (LCD)Step 1: Factor each denominator.Step 2: Find the least common denominator. The LCD is the product of all different factors from each denominator, with each factor raised to the greatest power that occurs in any denominator.

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7-2 Adding and Subtracting Rational Expressions

Adding or subtracting rational expressions with different denominators. Step 1: If the denominators are different, find the least common denominator. Write the rational expressions with the least common denominator and add or subtract numerators. Place the result over the common denominator.

Step 2: Simplify by writing all answers in lowest terms

• Example 2: Find the LCD for each pair of denominators

2

2

22

Solution: ( 6)

( 4)( 4), ( 4)

A) , 6

B)

Solut

16, 8 16

ion: ( - 4 )(

4)

x x

x

x x

x x x x x

x

x

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7-2 Adding and Subtracting Rational Expressions

Adding or subtracting rational expressions with different denominators. Example 3: Add or subtract as indicated

2 2 2

2 2

2

2

2( 3)

( 6

( 6) 6 2 6

( 6) ( 6) ( 6) ( 6)

( 4) ( 4)

( 4)( 4) ( 4)( 4)

( 4) ( 4) 4 4

( 4)( 4) ( 4)( 4) ( 4)( 4

1 1A) + =

6

1 1 B)

)

)

= 1

6 8 1

6

x x x x x

x x x x x x x x

x x

x

x x

x x x x x x

x x x x

x x x x x

x

x x

x

2

2Solution:

(

4)( 4)

x

x x

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7-2 Adding and Subtracting Rational Expressions

Adding or subtracting rational expressions with different denominators. Example 4: Add or subtract as indicated

2 22

2( 1) ( 2)( 3) ( 2)( 1) ( 2)( 1)

2 2 6 2 2

2 3A

6

( 2)( 1) ( 2)( 1) ( 2)( 1)

2 1 2 1

(

) - =2 1

2 1 B)

3) ( 3) ( 3)

4

( 2)( 1)

S

= 3

olution

3

x x x

x x x x

x x x x x x

x x x x x x

x

x x

x

x

x x

x x

x

x x

1

: ( 3)x

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7-2 Adding and Subtracting Rational ExpressionsExample 5: Add or subtract as indicated

2

2 2

2

4( ) 2( 5) 10 ( 5)( ) ( 5)( ) ( 5)( )

4 2 10 10 2

( 5)( ) ( 5)

4

( 4)( 1) ( 4)( 3)

( 3) 4 ( 1)

( 4)( 1)(

4 -2 10A) + =

5 x 5

4 B) =

3) (

3 4 7 12

5

)

( )

4

x

x x

x x x x x x

x x x

x x x x

x x

x x x x

x x

x x x

x x

x x x x

x x

x x x x

2 2 2

( 1)( 3)

3 4 4 5

( 4)( 1)( 3) ( 4)( 1)( 3)

( 5 1)

Solution: ( 4)( 1)( 3)

x x

x x x x x x

x x x x

x x

x x x

x x

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7-2 Adding and Subtracting Rational ExpressionsExample 6: Add or subtract as indicated

2

2

2

2

2

2

2

4 1

( 5)( 3)( 3)

4( 5) 1( 3)

( 3) ( 5) ( 3) ( 5)

4 20 3

( 3) ( 5

5 17

( 3) (

4 1 =

6

)

15

)

2

5

9 x xx

x x

x x x x

x

x x x x

x

x

x

x xx

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7-4 Equations with Rational Expressions and Graphs

Determining the domain of a rational equation.

The domain of a rational equation is the intersection (overlap)of the domains of the rational expressions in the equation.

• Example 1: Find the domain of the equation below:

3: ; 5 0 0

52

: ; 1 0 11

5: ; 1

Solution: x x 0

0

,

3 2

a

55

ll

1

real numbe

x

rs1

1

For x xx

For x xx

Fo

x x

r

21

7-4 Equations with Rational Expressions and Graphs

Solving rational equations.Multiply all terms of the equation by the least common denominator and solve. This technique may produce solutions that do not satisfy the original equation. All solutions have be checked in the original equation. Note: This technique cannot be used on rational expressions.

• Example 2:

-3x + 40 = 25 -3x = -15x = 5 Answer: {5}

Multiply all terms by 20x

3 2 520 20 20

2

3 2 5:

20 4

0 4

Solvex x

x x xx x

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7-4 Equations with Rational Expressions and Graphs

Solving rational equations.

Example 3:

2

1 2 3

1 ( 1)( 1) 1

Multiply all terms by ( 1)( 1)

1 2 3( 1)( 1)

1 2 3:

( 1)( 1) ( 1)( 1)1 ( 1)( 1) 1

Domain: { 1 11

1}

1(

No

t :

1

)

e

x x x x

x x

x x x x x xx

Solvex x

x

x

x x

x x

x

Since the domain does not include 1

2 3( 1) 1 2 3

Solution is the Null Set o

3

2 2 1

r

x x x

x x

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7-4 Equations with Rational Expressions and Graphs

Solving rational equations. Example 4:

2 2 2

4 1 2

( 3)( 2) ( 2)( 2) ( 3)( 2)

Multiply all terms by (x+3)( 2)

Domain: { 2, 3}

( 2)

4 1( 3)( 2)

4 1 2:

6 4 5 6

No

( 2) ( 3)( 2)( 2)( 3)( 2) ( )

t

2

e

2

:

( )

x x

Sol

x

v

x x x

x x

x x x

x x x x x

ex x x x x

xx x x x

2( 3)( 2)( 2)

( 3)( 2)

4( 2) 1( 3) 2

9 S

( 2) 4 8 3 2 4

olution: {-9}

x x xx x

x x x x

x

x x

24

7-4 Equations with Rational Expressions and Graphs

Solving rational equations. Example 5:

2

2

2 1 ( 3)

( 3) ( 1) ( 3)( 1)

Multiply all terms by (x+3)( 1)

2 1 ( 3)( 3)( 1) ( 3)( 1) ( 3)( 1)

( 3)

No Domain

2 1 3:

3

: { 1, 3

1

t

( 1) ( 3

2 3

)( )

e }

1

:

x x

x x

x x x

x x

x

x xx x x x x x

x x x

x xSolve

x x x

x

x

2

2

2( 1) 1( 3) ( 3) 2 2 3 3

4 5 0 ( 5)( 1) 0 since Solution: {-5 5 and }

1 1 x

x x x x x x x x

x x xx xx

25

7-4 Equations with Rational Expressions and Graphs

Recognizing the graph of a rational function.

Because one or more values of x are excluded from the domain of most rational functions, their graphs are usually discontinuous. The graph produces a vertical asymptote at the value of x that is not allowed as part of the domain.

• Example 6: Domai1

( ) n : 0} {f x x xx

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7-4 Equations with Rational Expressions and Graphs

Recognizing the graph of a rational function.

Because one or more values of x are excluded from the domain of most rational functions, their graphs are usually discontinuous. The graph produces a vertical asymptote at the value of x that is not allowed as part of the domain.

• Example 7:Do

2( ma) in:

3{ 3}f x x

xx