Transcript Adding/Subtracting Rational Expressions

Chapter 7
Section 4
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
7.4
1
2
3
Adding and Subtracting Rational
Expressions
Add rational expressions having the same
denominator.
Add rational expressions having different
denominators.
Subtract rational expressions.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
Add rational expressions having
the same denominator.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 3
Add rational expressions having the same
denominator.
We find the sum of two rational expressions with the same
procedure that we used in Section 1.1 for adding two fractions
having the same denominator.
P
R
If Q and Q (Q
≠ 0) are rational expressions, then
P R PR
 
.
Q Q
Q
That is, to add rational expressions with the same denominator,
add the numerators and keep the same denominator.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 4
EXAMPLE 1
Adding Rational Expressions
with the Same Denominator
Add. Write each answer in lowest terms.
Solution:
7 3
73


15 15
15
2x
2y

x y x y
10

15
25

35
2

3
2x  2 y 2 x  y


2
x y
x y
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 5
Objective 2
Add rational expressions having
different denominators.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 6
Add rational expressions having different
denominators.
We use the following steps, which are the same as those used
in Section 1.1 to add fractions having different denominators.
Step 1: Find the least common denominator (LCD).
Step 2: Rewrite 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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 7
EXAMPLE 2
Adding Rational Expressions
with Different Denominators
Add. Write each answer in lowest terms.
Solution:
1 1

10 15
1 3 1 2
3
2
   


10 3 15 2
30 30
5
3 2


30
30
1

6
10  2  5
15  3  5
LCD  2  3  5  30
m 2

3n 7 n
m 7 2 3
7m
6
7m  6
  
 


3n 7 7 n 3
21n 21n
21n
LCD  3  7  n  21n
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 8
EXAMPLE 3
Adding Rational Expressions
Add. Write the answer in lowest terms.
2
4p
 2
p 1 p 1
Solution:
2
4p
2 p  1
4p





p  1  p  1 p  1
p  1 p  1  p  1 p  1
2 p  2   4 p


 p  1 p  1
2p  2

 p  1 p  1
2  p  1

 p  1 p  1
2

p 1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 9
EXAMPLE 4
Adding Rational Expressions
Add. Write the answer in lowest terms.
2k
3
 2
2
k  5k  4 k  1
Solution:
2k
3
k  1
k  4


2k
3






 k  4 k 1  k 1 k  1  k  4 k 1  k  1  k 1 k  1  k  4
2k  k  1
3 k  4


 k  4 k 1 k  1  k  4 k 1 k  1
2k 2  5k  12

 k  4 k 1 k  1
2k 2  2k  3k  12

 k  4 k 1 k  1
2k  3 k  4


 k  4 k 1 k  1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 10
EXAMPLE 5
Adding Rational Expressions
with Denominators That Are
Opposites
Add. Write the answer in lowest terms.
m
n

2m  3n 3n  2m
Solution:
1

m
n
m
n





2m  3n 3n  2m
2m  3n 3n  2m  1
m
n


2m  3n 2m  3n
mn

2m  3n
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 11
Objective 3
Subtract rational expressions.
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Slide 7.4 - 12
Subtract rational expressions.
Use the following rule to subtract rational expressions
having the same denominator.
If
R
R
and (Q ≠ 0) are rational expressions, then
Q
Q
P R PR


Q Q
Q
That is, to subtract rational expressions with the same
denominator, subtract the numerators and keep the same
denominator.
We subtract rational expressions having different
denominators using a procedure similar to the one used to
add rational expressions having different denominators.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 13
EXAMPLE 6
Subtracting Rational
Expressions with the Same
Denominator
Subtract. Write the answer in lowest terms.
5t 5  t

t 1 t 1
Solution:
5t   5  t 

t 1
5t  5  t

t 1
4t  5

t 1
Sign errors often occur in subtraction problems. The numerator of
the fraction being subtracted must be treated as a single quantity.
Be sure to use parentheses after the subtraction sign.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7.4 - 14
EXAMPLE 7
Subtracting Rational
Expressions with Different
Denominators
Subtract. Write the answer in lowest terms.
6
1

a  2 a 3
Solution:
6a  18
a2
6 a 3 1 a  2






a  2 a 3 a 3 a  2
 a  2 a  3  a  3 a  2
6a 18   a  2


 a  2 a  3
6a  18  a  2

 a  2 a  3
5a  20

 a  2 a  3
5  a  4

 a  2 a  3
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Slide 7.4 - 15
EXAMPLE 8
Subtracting Rational
Expressions with Denominators
That Are Opposite
Subtract. Write the answer in lowest terms.
4 x 3 x  1

x 1 1 x
Solution:
4 x 3x  1  1
4 x 3x  1





x  1 1  x  1
x 1 x 1
4 x  3x  1

x 1
x 1

x 1
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
4 x   3x  1

x 1
1
Slide 7.4 - 16
EXAMPLE 9
Subtracting Rational
Expressions
Subtract. Write the answer in lowest terms.
3r
4

r 2  5r r 2  10r  25
Solution:
3r
4


r  r  5  r  5 r  5
3r
r 5
4
r




r  r  5 r  5  r  5 r  5 r
3r 2 15r
4r


r  r  5 r  5 r  r  5 r  5
3r 2  19r

r  r  5 r  5
r  3r  19 

r  r  5 r  5
3r  19 


2
 r  5
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Slide 7.4 - 17