Section 14.4 Adding and Subtracting Rational Expressions

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. Add rational expressions having the same
denominator.
2. Add rational expressions having different
denominators.
3. Subtract rational expressions.
Objectives
14.4 Adding and Subtracting 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
Add Rational Expressions Having the Same Denominator
Adding Rational Expressions (Same Denominator)
The rational expressions and (where Q ≠ 0) are
added as follows.
In words: To add rational expressions with the same
denominator, add the numerators and keep the same
denominator.
P
Q
R
Q
P R P R
Q Q Q

 

4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write each answer in lowest terms.
7 8
(a)
18 18

Add Rational Expressions Having the Same Denominator
7 8
18


15
18

5
6

2 2
2 2
6 4
(b)
1 1
 

 
x x x x
x x
2 2
2
6 4
1
  


x x x x
x
2
2
5 5
1
x x
x



5
( 1
( 1)
( 1) )
x
x
x
x



5
1
x
x


6
3
3
5



5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Add Rational Expressions Having Different Denominators
Adding Rational Expressions (Different Denominators)
Step 1 Find the least common denominator (LCD).
Step 2 Write each rational expression as an
equivalent rational expression with the LCD as
the denominator.
Step 3 Add the numerators to get the numerator of the
sum. The LCD is the denominator of the sum.
Step 4 Write in lowest terms using the fundamental
property.

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write the answer in lowest terms.
5 4
6 15

Add Rational Expressions Having Different Denominators
25 8
30 30
 
33
30

11 1
, or 1
10 10


7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write the answer in lowest terms.
2
2
3 2
16 4
a a
a a


 
Add Rational Expressions Having Different Denominators
  
2
3 2
4 4 4
a a
a a a

 
  
  
 
  
2
2 43
4 4 4 4
a aa
a a a a
 
 
   
     
2 2
3 2 8
4 4 4 4
a a a
a a a a
 
 
   
  
2
8
4 4
a a
a a


 

8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write the answer in lowest terms.
2 2
4 3
8 15 3 10
z z
z z z z


   
Add Rational Expressions Having Different Denominators
4 3
( 3)( 5) ( 2)( 5)
z z
z z z z

 
   
4 ( 3)
( 3)( 5) ( 2)(
( 2) ( 3)
( 2) 35)( )
z zz z
z zz z z z

 
   
 

2 2
4 8 9
( 3)( 5)( 2)
z z z
z z z
  

  
2
5 8 9
( 3)( 5)( 2)
z z
z z z
 

  

9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Add. Write the answer in lowest terms.
Add Rational Expressions Having Different Denominators
2
5 7
1 1
y
y y


 
2
5 7
1 ( 1)1
y
y y


 
 
2
5 7
1 1
y
y y

 

 
2
5 7
1
y
y
 


2
12
1
y
y




10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Subtract Rational Expressions
Subtracting Rational Expressions (Same Denominator)
The rational expressions and (where Q ≠ 0) are
subtracted as follows.
In words: To subtract rational expressions with the same
denominator, subtract the numerators and keep the same
denominator.
P
Q
R
Q
.
P R P R
Q Q Q

 

11. Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Subtract. Write the answer in lowest terms.
3 1 2 5
2 2
x x
x x
 

 
Subtract Rational Expressions
3 1 (2 5)
2
x x
x
  


3 1 2 5
2
x x
x
  


4
2
x
x




12. Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Subtract Rational Expressions
CAUTION
Sign errors often occur in problems like the previous
one. The numerator of the fraction being subtracted
must be treated as a single quantity.
Be sure to use parentheses after the subtraction
sign.

13. Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Subtract. Write the answer in lowest terms.
2 3 2
3 5 1x x
x x x



Subtract Rational Expressions
2 2
3 5 1
( 1)
x x
x x x

 

2 2
3 (5 1)
( 1
( 1
)
)
( 1)
x
x
x x
x x x

 




2
2
3 3 5 1
( 1)
x x x
x x
  


2
2
3 8 1
( 1)
x x
x x
 



14. Slide - 14Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Subtract. Write the answer in lowest terms.
2 2
3 9
2 7 4 7 12x x x x

   
Subtract Rational Expressions
3 9
(2 1)( 4) ( 3)( 4)x x x x
 
   

3(x  3)
(2x 1)(x  4)(x  3)

9(2x 1)
(x  3)(x  4)(2x 1)

3x  9
(2x 1)(x  4)(x  3)

18x  9
(x  3)(x  4)(2x 1)

3x  9 18x  9
(2x 1)(x  4)(x  3)

15x 18
(2x 1)(x  4)(x  3)