Chapter 7 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter Chapter 77Section Section 11

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


The Fundamental Property of Rational Expressions

Find the numerical value of a rational expression.Find the values of the variable for which a rational expression is undefined.Write rational expressions in lowest terms.Recognize equivalent forms of rational expressions.

11

44

33

22

7.17.17.17.1


P

QA rational expression is an expression of the form , where P and Q are polynomials, with Q ≠ 0.

and9

,3

x

y

32

8

m3

6

8,

x

x

Examples of rational expressions

The Fundamental Property of Rational Expressions

The quotient of two integers (with the denominator not 0), such as or , is called a rational number. In the same way, the quotient of two polynomials with the denominator not equal to 0 is called a rational expression.

Slide 7.1 - 3

2

3

3

4


Objective 11

Find the numerical value of a rational expression.

Slide 7.1 - 4


EXAMPLE 1

Find the value of the rational expression, when x = 3.

Solution:

Evaluating Rational Expressions

Slide 7.1 - 5

2 1

x

x

1

3

2 3

3

6 1

3

7


Objective 22

Find the values of the variable for which a rational expression is undefined.

Slide 7.1 - 6

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7.1 - 7

Find the values of the variable for which a rational expression is undefined.

3 6

2 4

x

x

For instance, in the rational expression

the variable x can take on any real number value except 2. If x is 2, then the denominator becomes 2(2) − 4 = 0, making the expression undefined. Thus, x cannot equal 2. We indicate this restriction by writing x ≠ 2.

In the definition of a rational expression , Q cannot equal 0. The denominator of a rational expression cannot equal 0 because division by 0 is undefined.

P

Q

Since we are solving to find values that make the expression undefined, we write the answer as “variable ≠ value”, not “variable = value or { } .

Denominator cannot equal 0

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7.1 - 8

Find the values of the variable for which a rational expression is undefined. (cont’d)

To determine the values for which a rational expression is undefined, use the following procedure.

Step 1: Set the denominator of the rational expression equal to 0.

Step 2: Solve this equation.

Step 3: The solutions of the equation are the values that make the rational expression undefined.

The numerator of a rational expression may be any real number. If the numerator equals 0 and the denominator does not equal 0, then the rational expression equals 0.


Find any values of the variable for which each rational expression is undefined.

EXAMPLE 2

Solution:

Finding Values That Make Rational Expressions Undefined

Slide 7.1 - 9

2

5

x

x

2

3

6 8

r

r r

2

5 1

5

z

z

5 0x

2 0r

5 55 0x 5x

3

2 4

r

r r

2 22 0r

2r

4 0r 4 44 0r

4r

never undefined


Objective 33

Slide 7.1 - 10

Write rational expressions in lowest terms.


A fraction such as is said to be in lowest terms. The idea of greatest common factor is used for this definition, which applies to all rational expressions.

A rational expression (Q ≠ 0) is in lowest terms if the greatest common factor of its numerator and denominator is 1.

Write rational expressions in lowest terms.

Slide 7.1 - 11

2

3

.PK P

QK Q

If (Q ≠ 0) is a rational expression and if K represents any polynomial, where K ≠ 0, then

P

Q

P

Q

1 .K K

K K

P P P P

Q Q Q Q

This property is based on the identity property of multiplication, since


EXAMPLE 3

Solution:

3

2

6

2

p

p

Writing in Lowest Terms

Slide 7.1 - 12

Write each rational expression in lowest terms.

15

45

32

2

p pp

p p

3 5

3 53

3p

1

3


To write a rational expression in lowest terms, follow these steps.

Step 1: Factor the numerator and denominator completely.

Write rational expressions in lowest terms. (cont’d)

Slide 7.1 - 13

Step 2: Use the fundamental property to divide out any common factors.

Rational expressions cannot be written in lowest terms until after the numerator and denominator have been factored. Only common factors can be divided out, not common terms.

2 3

2

36 9 3

4 6 2 23

x

x

x

x

6

4

x

x

Numerator cannot be factored.

If the numerator and the denominator of a rational expression are opposites, as in , then the rational expression is equal to −1.

x y

y x


EXAMPLE 4

Solution:2

3

2 12

3 2 1

y

y

4 2

6 3

y

y

a ba b

a a bb

a b

a b

Writing in Lowest Terms

Slide 7.1 - 14

Write in lowest terms.

2 2

2 22

a b

a ab b


EXAMPLE 5

Write in lowest terms.2

2

5

5

z

z

Writing in Lowest Terms (Factors Are Opposites)

Slide 7.1 - 15

Solution:

2

2

5

5

1

1

z

z

1


Write each rational expression in lowest terms.

EXAMPLE 6

5

5

y

y

Writing in Lowest Terms (Factors are Opposites)

Slide 7.1 - 16

225 16

12 15

x

x

9

9

k

k

5

5

1 y

y

5 4

3

5 4

5 4

xx

x

already in lowest terms

5 4

3

k

1

5 4

3

k or


Objective 44

Slide 7.1 - 17

Recognize equivalent forms of rational expressions.


Recognize equivalent forms of rational expressions.

Slide 7.1 - 18

The − sign representing the factor −1 is in front of the expression, even with fraction bar. The factor −1 may instead be placed in the numerator or in the denominator. Some other equivalent forms of this rational expression are

and

When working with rational expressions, it is important to be able to recognize equivalent forms of an expressions. For example, the common fraction can also be written and . Consider the rational expression

.

2 3

2

x

5

65

6

5

6

2 3

2

x 2 3

2

x


Recognize equivalent forms of rational expressions. (cont’d)

Slide 7.1 - 19

By the distributive property,

can also be written 2 3

2

x

is not an equivalent form of . The sign preceding 3 in the numerator of should be − rather than +. Be careful to

apply the distributive property correctly.

2 3

2

x 2 3

2

x

2 3

2

x

2 3

2

x


EXAMPLE 7 Writing Equivalent Forms of a Rational Expression

Slide 7.1 - 20

Write four equivalent forms of the rational expression.

2 6

3

x

x

2 6

3

x

x

Solution:

2 6

3

x

x

2 6

3

x

x

2 6

3

x

x

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Chapter 7 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley