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Section 2 Chapter 7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
1
Objectives
2
3
Adding and Subtracting Rational Expressions
Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational expressions with different denominators.
7.2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add and subtract rational expressions with the same denominator.
Objective 1
Slide 7.2- 3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Adding or Subtracting Rational Expressions
Step 1 If the denominators are the same, add or subtract the numerators. Place the result over the common denominator.
If the denominators are different, first find the least common denominator. Write all rational expressions with this least common denominator, and then add or subtract the numerators. Place the result over the common denominator.
Step 2 Simplify. Write all answers in lowest terms.
Slide 7.2- 4
Add and subtract rational expressions with the same denominator.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add or subtract as indicated.
79 9x y+
79x y+
=
3 3
8 143 3x x
− 3
8 143x−
= 3
63x−
= 3
2x−
=
2 2 2 2
r tr t r t
+− − 2 2
r tr t+
=− ( )( )
r tr t r t
+=
+ −
1r t
=−
2 2
63 18 3 18
xx x x x
++ − + − 2
63 18x
x x+
=+ −
( )( )66 3x
x x+
=+ −
13x
=−
Slide 7.2- 5
CLASSROOM EXAMPLE 1 Adding and Subtracting Rational Expressions Same Denominator
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find a least common denominator.
Objective 2
Slide 7.2- 6
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Finding the Least Common Denominator
Step 1 Factor each denominator. Step 2 Find the least common denominator. The LCD is the
product of all of the different factors from each denominator, with each factor raised to the greatest power that occurs in any denominator.
Slide 7.2- 7
Find a least common denominator.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find the LCD for each group of denominators. 3 5 2 610 ,15a b a b
z, z + 6
Factor. 10a3b5 = 2 • 5 • a3 • b5
15a2b6 = 3 • 5 • a2 • b6
LCD = 2 • 3 • 5 • a3 • b6 = 30a3b6
Each denominator is already factored.
LCD = z(z + 6)
Slide 7.2- 8
CLASSROOM EXAMPLE 2 Finding Least Common Denominators
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Find the LCD for each group of denominators. m2 – 16 , m2 + 8m + 16
Factor. = (m +4)(m – 4)
LCD
m2 – 16
m2 + 8m + 16 = (m +4)2
= (m +4)2(m – 4)
x2 – 2x + 1, x2 – 4x + 3, 4x – 4
x2 – 2x + 1 = (x – 1)(x – 1)
x2 – 4x + 3 = (x – 1)(x – 3)
4x – 4 = 4(x – 1)
LCD = 4(x – 1)2(x – 3) Slide 7.2- 9
CLASSROOM EXAMPLE 2 Finding Least Common Denominators (cont’d)
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add and subtract rational expressions with different denominators.
Objective 3
Slide 7.2- 10
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add or subtract as indicated.
6 14m m
+
2 14y y
−+
6 14
44m m⋅
= +⋅
24 14m+
=254m
=
( )( )
( )( )
44
2 14y y yyyy
= −+
+
+
( )( ) ( )2 4
4 4y y
y y y y+
= −+ +
( )2 8
4y yy y+ −
=+
Slide 7.2- 11
CLASSROOM EXAMPLE 3 Adding and Subtracting Rational Expressions (Different Denominators)
Solution:
24 14 4m m
= +
( )84
yy y
+=
+
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Subtract.
5 7 142 7 2 7x xx x+ − −
−+ +
The denominators are already the same for both rational expressions. The subtraction sign must be applied to both terms in the numerator of the second rational expression.
15 72 7( 4)xxx
−=
− −+
+5 7 1
74
2x x
x+ ++
=+
6 212 7xx+
=+
( )2 732 7xx+
=+
3=
Slide 7.2- 12
CLASSROOM EXAMPLE 4 Subtracting Rational Expressions
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
2 32 1r
r r+
−− −
The LCD is (r – 2)(r – 1).
( )( )( )
( )( )( )( )1
221
1 232
rr
r rr rr− −
−
+=
− − −−
( )( )( )
22 22 1
6r rrr r−
=−
− +
−
−
( )( )
22 22 1
6r rrr r−
=−
− −
−
+
( )( )
2 42 1r rr r− + +
=− −
Slide 7.2- 13
CLASSROOM EXAMPLE 4 Subtracting Rational Expressions (cont’d)
Subtract.
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add.
2 1 .3 3x x+
− −
To get a common denominator of x – 3, multiply both the numerator and denominator of the second expression by –1.
( )( )( )1 12
3 3 1x x−
= +− − −
2 13 3x x
−= +
− −
( )2 13x
+ −=
−
13x
=−
Slide 7.2- 14
CLASSROOM EXAMPLE 5 Adding and Subtracting Rational Expressions (Denominators Are Opposites)
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add and subtract as indicated.
2
4 2 105 5x x x x
−+ −
− −
( )4 2 105 5x x x x
−= + −
− −
( )( )( ) ( )
524 15 55
0xxxx x xx x
−= −
−
−+
− −
( )( )( )
4 2 5 105
x xx x
+ − − −=
−
( )4 2 10 10
5x xx x
+ − + −=
− ( )25x
x x=
−
25x
=−
Slide 7.2- 15
CLASSROOM EXAMPLE 6 Adding and Subtracting Three Rational Expressions
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Subtract.
2 2
43 4 7 12a a
a a a a−
−+ − + +
( )( ) ( )( )4
4 1 4 3a a
a a a a−
= −+ − + +
( )( )( )( )
34 1 3a a
a a a− +
=+ − +
LCD is ( )( )( )4 1 3 .a a a+ − +
( )( )( )( )
4 14 1 3a a
a a a−
−+ − +
Slide 7.2- 16
CLASSROOM EXAMPLE 7 Subtracting Rational Expressions
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
( ) ( )( )( )( )
3 4 14 1 3
a a a aa a a− + − −
=+ − +
( )( )( )
2 23 4 44 1 3
a a a aa a a− − − +
=+ − +
( )( )( )
254 1 3
a aa a a
− +=
+ − +
Distributive property
Combine terms in the numerator.
Slide 7.2- 17
CLASSROOM EXAMPLE 7 Subtracting Rational Expressions (cont’d)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Add.
2 2
4 16 9 2 15p p p p
+− + + −
( )( ) ( )
( )( )( )2 2
4 5 1 33 5 5 3p p
p p p p+ −
= +− + + −
LCD is
( )( ) ( )( )4 13 3 5 3p p p p
= +− − + −
( ) ( )23 5 .p p− +
Slide 7.2- 18
CLASSROOM EXAMPLE 8 Adding Rational Expressions
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
( ) ( )( ) ( )2
4 5 1 33 5
p pp p+ + −
=− +
( ) ( )24 20 3
3 5p pp p+ + −
=− +
( ) ( )25 173 5p
p p+
=− +
Distributive property
Combine terms in the numerator.
Slide 7.2- 19
CLASSROOM EXAMPLE 8 Adding Rational Expressions (cont’d)
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