chapter 7 section 1. copyright © 2012, 2008, 2004 pearson education, inc. objectives 1 the...

Chapter 7 Section 1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Objectives

1

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.

7.1

2

3

4


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

,P

Q

9,

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.

2

3

3,

4

Slide 7.1-3


Objective 1

Find the numerical value of a rational expression.

Slide 7.1-4


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

Solution:

2 1

x

x

1

3

2 3

3

6 1

3

7

Slide 7.1-5

EXAMPLE 1 Evaluating Rational Expressions


Objective 2


Slide 7.1-6



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 { } .

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.

3 6

2 4

x

x

Denominator cannot equal 0

Slide 7.1-7

Ѳ


Determining When a Rational Expression is Undefined

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 variable cannot equal these values.

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.

Slide 7.1-8

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


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

Solution:

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

Slide 7.1-9

EXAMPLE 2 Finding Values That Make Rational Expressions Undefined

Ѳ


Objective 3


Slide 7.1-10


A fraction such as is said to be in lowest terms.


2

3

Fundamental Property of Rational ExpressionsIf (Q ≠ 0) is a rational expression and if K represents any

polynomial, where K ≠ 0, then .

PK P

QK Q

P

Q

Lowest TermsA rational expression (Q ≠ 0) is in lowest terms if the greatest

common factor of its numerator and denominator is 1.

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

Slide 7.1-11


Solution:

3

2

6

2

p

p

Write each rational expression in lowest terms.

15

45

32

2

p pp

p p

3 5

3 53

3p

1

3

Slide 7.1-12

EXAMPLE 3 Writing in Lowest Terms


Quotient of Opposites

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

Writing a Rational Expression in Lowest Terms

Step 1: Factor the numerator and denominator completely.

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.

Slide 7.1-13

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


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

Write in lowest terms.

2 2

2 22

a b

a ab b

Slide 7.1-14

EXAMPLE 4 Writing in Lowest Terms


Write in lowest terms.2

2

5

5

z

z

Solution:

2

2

5

5

1

1

z

z

1

Slide 7.1-15

EXAMPLE 5 Writing in Lowest Terms (Factors Are Opposites)


Write each rational expression in lowest terms.

5

5

y

y

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

x

1

5 4

3

x or

Slide 7.1-16

EXAMPLE 6 Writing in Lowest Terms (Factors Are Opposites)

Solution:


Objective 4


Slide 7.1-17



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 expression2 3

.2

x

5.

65

6

5

6

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 2 3

2

x 2 3

2

x

Slide 7.1-18


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

By the distributive property,

can also be written 2 3

2

x 2 3.

2

x

Slide 7.1-19

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


Write four equivalent forms of the rational expression.

2 7

3

x

x

2 7,

3

x

x

Solution:

2 7

,3

x

x

2 7,

3

x

x

2 7

3

x

x

Slide 7.1-20

EXAMPLE 7 Writing Equivalent Forms of a Rational Expression


HL 7.1 Book Beginning Algebra Page 426 Exercises 3,4,5,18,20,23,24,27,34,36,39.Page 427 Exercises 40,41,42,46,47,48,49,54,61,62,67.

chapter 7 section 1. copyright © 2012, 2008, 2004 pearson education, inc. objectives 1 the...

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