Chapter 7 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
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Transcript of Chapter 7 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Chapter Chapter 77Section Section 22
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
11
22
33
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.
7.27.27.27.2
Slide 7.2- 3Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
Add and subtract rational expressions with the same denominator.
Slide 7.2- 4Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Adding or Subtracting Rational ExpressionsStep 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- 5Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2 2
6
3 18 3 18
x
x x x x
2 2 2 2
r t
r t r t
3 3
8 14
3 3x x
7
9 9
x y
Add or subtract as indicated.
EXAMPLE 1
Slide 7.2- 6Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 2
Find a least common denominator.
Slide 7.2- 7Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Finding the Least Common DenominatorStep 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- 8Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2
Find the LCD for each group of denominators.
a. 3 5 2 610 ,15a b a b
b. z, z + 6
Factor. 10a3b5 = 2 • 5 • a3 • b5
15a2b6 = 3 • 5 • a2 • b6
LCD = 2 • 3 • 5 • a3 • b5 = 30a3b6
Each denominator is already factored.
LCD = z(z + 6)
Slide 7.2- 9Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
c. m2 – 16 , m2 + 8m + 16
Factor. = (m +4)(m – 4)
LCD
m2 – 16
m2 + 8m + 16 = (m +4)2
= (m +4)2(m – 4)
d. 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- 10Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 3
Add and subtract rational expressions with different denominators.
Slide 7.2- 11Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Add or subtract as indicated.
a. 6 1
4m m
b. 2 1
4y y
6 1
4
4
4m m
24 1
4 4m m
24 1
4m
25
4m
4
4
2 1
4y y y
yy
y
2 4
4 4
y y
y y y y
2 8
4
y y
y y
8
4
y
y y
Example 3
Slide 7.2- 12Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Subtract.
a. 5 7 14
2 7 2 7
x x
x 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 7
2 7
( 4)xx
x
5 7 1
7
4
2
x x
x
6 21
2 7
x
x
2 73
2 7
x
x
3
Slide 7.2- 13Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2 3
2 1
r
r r
continued
b.
Slide 7.2- 14Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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 1
3 3x x
2 1
3x
1
3x
The equivalent answer is
1 1
1 3 x
1.
3x
Example 5
Slide 7.2- 15Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Simplify.
Example 6
x
x
xx
365
32
Slide 7.2- 16Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Simplify
Example 7
2
4 2 10
5 5x x x x
Slide 7.2- 17Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Simplify.
Example 8
Slide 7.2- 18Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Simplify.
Example 9
45
3
56
422
xxxx
Slide 7.2- 19Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Subtract
Example 10
2 2
4
3 4 7 12
a a
a a a a
Slide 7.2- 20Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Add
Example 11
2 2
4 1
6 9 2 15p p p p