This document discusses rational expressions and their applications. It covers objectives like finding the numerical value of a rational expression, determining values that make a rational expression undefined, writing rational expressions in lowest terms, and recognizing equivalent forms. Some key points include the definition of a rational expression as P/Q where P and Q are polynomials and Q ≠ 0, setting the denominator equal to 0 to find undefined values, and using the fundamental property of rational expressions to write expressions in lowest terms by dividing out common factors. Examples are provided for each concept.

1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Rational
Expressions and
Applications
14

2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
1. Find the numerical value of a rational
expression.
2. Find the values of the variable for which a
rational expression is undefined.
3. Write rational expressions in lowest terms.
4. Recognize equivalent forms of rational
expressions.
Objectives
14.1 The Fundamental Property of Rational
Expressions

3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Rational Expression
A rational expression is an expression of the form
where P and Q are polynomials, with Q ≠ 0.
,
P
Q

4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Find the Numerical Value of a Rational Expression
Example
Find the numerical value of for each value of z.2
2 7
36
z
z


(a) z = –1
2 2
2 7 2( ) 7
36 ( 36
1
1)
z
z


 

 
2 7
1 36
 


5
35


1
7
 
(b) z = 6
2 2
2 7 2(6) 7
36 ( ) 36 6
z
z
 

 
12 7
36 36



0
19

The expression is undefined
when z = 6.

5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
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.
Find the Values of the Variable for Which a
Rational Expression is Undefined

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Find the Values of the Variable for Which a
Rational Expression is Undefined
Example
Find any values of the variable for which each rational
expression is undefined.
2
3
(b)
10 9
x
x x

 
Set the denominator equal to 0.x2 – 10x + 9 = 0
(x – 1)(x – 9) = 0
x – 1 = 0 or x – 9 = 0
x = 1 x = 9
The original expression is undefined for x = 1 or x = 9.
Factor.
Zero-factor property
Solve.

7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Find the Values of the Variable for Which a
Rational Expression is Undefined
Example (cont)
Find any values of the variable for which each rational
expression is undefined.
3
2
(c)
25
y
y 
Set the denominator equal to 0 and solve.
y2 + 25= 0
Thus, there are no values for which this rational expression
is undefined.
The denominator will not equal 0 for any value of y because y2
is always greater than or equal to 0, and adding 25 makes the
sum greater than 0.

8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Lowest Terms
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
Fundamental Property of Rational Expressions
If (Q ≠ 0) is a rational expression and if K represents
any polynomial, (where K ≠ 0), then the following holds
true.
P
Q

PK P
QK Q
P
Q

9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
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.
Write Rational Expressions in Lowest Terms

10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
8 24
(a)
10 30
x
x


Example
Write each rational expression in lowest terms.
Write Rational Expressions in Lowest Terms
8
10
( 3)
( 3)
x
x



8
10

4
5

2
2
3 6
(b)
2
y y
y y

 
( 2)
( 21 )
3
( )
y
y
y
y




3
1
y
y



11. Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
CAUTION
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. For example,
Write Rational Expressions in Lowest Terms
( )
( )



 

x
x
x
x
6 9 2 3
2 3
3 3
4 6 2 2
Divide out the
common factor.
6
4
x
x
 Numerator cannot
be factored.
Already in
lowest terms.

12. Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
x y
y x


Example
Write the rational expression in lowest terms.
Write Rational Expressions in Lowest Terms
At first, it does not appear that the numerator
and denominator share any common factor.
However, use the fact that the denominator, y – x, can be
rewritten as
y – x = –1(–y +x) = –1(x – y)
Thus,
1( )
x y x y
y x x y
 

  
1
1


1 

13. Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
If the numerator and denominator of a rational expression
are opposites, such as in then the rational
expression is equal to –1.
Write Rational Expressions in Lowest Terms
,


x y
y x
CAUTION
Although x and y appear in both the numerator and
denominator in the previous example, we cannot use the
fundamental property right away because they are terms,
not factors. Terms are added, while factors are
multiplied.

14. Slide - 14Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
2 5
x
x


Example
Write three equivalent forms of the rational expression.
Recognize Equivalent Forms of Rational Expressions
Consider different placements for the – sign.
2
1.
2 5
x
x


2 2
3.
2 5 5 2
x x
x x  

2
2.
(2 5)
x
x 