Transcript 6.2 - Web4students
Copyright © 2012 Pearson Education, Inc.
Slide 7- 1
6.2
Rational Expressions and Functions: Adding and Subtracting
■ ■ When Denominators Are the Same When Denominators Are Different
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Addition and Subtraction with Like Denominators
To add or subtract when the denominators are the same, add or subtract the numerators and keep the same denominator.
A C
B C
A
C B
, and
A C
B C
C
, where
C
0.
Copyright © 2012 Pearson Education, Inc.
Slide 6- 3
Example
Add. Simplify the result, if possible.
a) 5 6
w
b)
w w x
4
x
7 c) 3
x
2 3
x
4
x
1 9
x
2 3
x
1 3 d)
x
2
x
9 36 3
x x
x
2 7 2 3 36
Solution
a) 5 6
w w w
11
w w
The denominators are alike, so we add the numerators.
b) 4
x
x
7 3
x
2
x
7 7
x x
7 2 The denominators are alike, so we add the numerators.
Slide 6- 4
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Example continued
c) 3
x
2 4
x
3
x
1 9
x
2 3
x
1 3 (3
x
2 4
x
3
x
1
x
2 3 ) d)
x
2
x
9 36
x
2 3 36 (
x
2 4
x
2 5
x
12 3
x
1
x
6 36
x
6 Combining like terms Combining like terms in the numerator
x
6)(
x
6) Factoring
x
6) (
x
6) (
x
6)
x
1 6
Slide 6- 5
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Example
Subtract and, if possible, simplify: a)
x
5
x
3
x x
4 3 b)
x x
2 6
x x
30 6
Solution
a)
x
5
x
3
x
4
x
3 5
x
(
x x
3 4 ) 5
x x
3 4
x
x
3 4 4 The parentheses are needed to make sure that we subtract both terms.
Removing the parentheses and changing the signs (using the distributive law) Combining like terms
Slide 6- 6
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Example continued
b)
x x
2 6
x
30
x
6
x
2 (
x
30 )
x
6
x
2
x
6 30 (
x
6)(
x
5)
x
6 Removing the parentheses (using the distributive law) Factoring, in hopes of simplifying (
x
6) (
x
5)
x
6 Removing a factor equal to 1 5
Slide 6- 7
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Least Common Multiples and Denominators
To add or subtract rational expressions that have different denominators, we must first find equivalent rational expressions that
do
have a common denominator.
The least common multiple (LCM) must include the factors of each number, so it must include each prime factor the greatest number of times that it appears in any factorizations.
Slide 6- 8
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Example
For each pair of polynomials, find the least common multiple.
a) 16
a
and 24
b
b) 24
x
4
y
4 c)
x
2 and 6 4 and
x
2
x
6
y
2 2
x
8
Solution
a) 16 24
a
=
b =
2 2 2 2 2 2 The LCM = 2 3 2
a b
2 2 2
a
3
b
16
a
is a factor of the LCM 24
b
is a factor of the LCM The LCM is 2 4 3
a
b
, or 48
ab
Slide 6- 9
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Example continued
b) 24
x
4
y
4 6
x
6
y
2 LCM = 2 = 2 = 2 2 2 2 3 3 2
x
x
3
x
x x x
x x x
x x x
y
x x
y
y y
y
y y
y
y x
y x
Note that we used the highest power of each factor. The LCM is 24
x
6
y
4 c)
x
2
x
2 4 = 2
x
(
x
8 = ( 2)(
x x
+ 2) + 2)(
x
4) LCM = (
x
2)(
x
+ 2) (
x
4)
x
2 4 is a factor of the LCM
x
2 2
x
8 is a factor of the LCM
Slide 6- 10
Copyright © 2012 Pearson Education, Inc.
Example
For each group of polynomials, find the least common multiple.
a) 15
x
, 30
y
, 25
xyz
b)
x
2 + 3,
x
+ 2, 7
Solution
a) 15
x
= 3 30
y =
2 25
xyz
= 5 5 3 5
x
5
x
LCM = 2 3
y
y
5
z
5 The LCM is 2 3 5 2
x
x y
y z
z
or 150
xyz
b) Since
x
2 + 3,
x
+ 2, and 7 are not factorable, the LCM is their product: 7(
x
2 + 3)(
x
+ 2).
Slide 6- 11
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To Add or Subtract Rational Expressions
1.
2.
Determine the
least common denominator
(LCD) by finding the least common multiple of the denominators.
Rewrite each of the original rational expressions, as needed, in an equivalent form that has the LCD.
3.
Add or subtract the resulting rational expressions, as indicated.
4.
Simplify the result, if possible, and list any restrictions, on the domain of functions.
Slide 6- 12
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Example
Add: 4
x
2 9 5
x
.
12
Solution
1.
First, we find the LCD: 9 = 3 12 = 2 3 2 3 LCD = 2 2 3 3 = 36
2.
Multiply each expression by the appropriate number to get the LCD.
4
x
2 9 5
x
12 4
x
2 4
x
2 4 5
x
5
x
3 3 16
x
2 36 15
x
36 Copyright © 2012 Pearson Education, Inc.
Slide 6- 13
Example continued
3.
Next we add the numerators: 16
x
36 2 15 36
x
16
x
2 36 15
x
.
4.
Since 16
x
2 + 15
x
and 36 have no common factor, 16
x
2 15
x
cannot be simplified any further. 36 Subtraction is performed in much the same way.
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Slide 6- 14
Example
Subtract: 7 9
x
5 12
x
2 .
Solution
We follow the four steps as shown in the previous example. First, we find the LCD.
9
x
= 3 3
x
12
x
2 = 2 2 3
x
x
LCD = 2 2 3 3
x
x
= 36
x
2 The denominator 9
x
must be multiplied by 4
x
to obtain the LCD.
The denominator 12
x
2 must be multiplied by 3 to obtain the LCD.
Slide 6- 15
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Example continued
Multiply to obtain the LCD and then we subtract and, if possible, simplify.
7 9
x
5 12
x
2 7 9
x
4
x
4
x
5 12
x
2 3 3 28
x
36
x
2 15 36
x
2 28
x
15 36
x
2
Caution!
Do not simplify
these
rational expressions or you will lose the LCD.
This cannot be simplified, so we are done.
Slide 6- 16
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Example
Add:
a
2 3
a
4
a
2 2 2
a
.
Solution
First, we find the LCD:
a
2 4 = (
a
2)(
a
+ 2)
a
2 2
a
=
a
(
a
2) LCD =
a
(
a
2)(
a
+ 2).
We multiply by a form of 1 to get the LCD in each expression:
a
2 3
a
4
a
2 2 2
a
(
a
3
a
2)(
a
2)
a a
(
a
2 2 )
a a
2 2 Copyright © 2012 Pearson Education, Inc.
Slide 6- 17
Example continued
a
2 3
a
4
a
2 2 2
a
3
a
(
a
2)(
a
2)
a
3
a
2 2)(
a
2) 3
a
2 2
a
2)(
a
4 2) 2
a
(
a
2 )
a a
2 2 2
a
4 2)(
a
2) 3
a
2 + 2
a
+ 4 will not factor so we are done.
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Slide 6- 18
Example
Subtract:
x
2
x
4
x x
1 6 .
Solution
First, we find the LCD. It is just the product of the denominators: LCD = (
x
+ 4)(
x
+ 6).
We multiply by a form of 1 to get the LCD in each expression. Then we subtract and try to simplify.
x
2
x
4
x
1
x
6
x
x
2 4
x
6
x
6
x x
1 6
x
4
x
4
x
2 (
x
8
x
12 4)(
x
6) (
x x
2 3
x
4)(
x
4 6) Multiplying out numerators
Slide 6- 19
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Example continued
x
2 (
x
8
x
12 4)(
x
6) (
x x
2 3
x
4)(
x
4 6)
x
2 8
x
12 (
x
( 4)(
x x
2 3
x
6) 4 )
x
2 8
x
(
x
12 4) (
x x
2 6) 3
x
4 (
x
5
x
4)( 16
x
6) When subtracting a numerator with more than one term, parentheses are important.
Removing parentheses and subtracting every term
Slide 6- 20
Copyright © 2012 Pearson Education, Inc.
Example
Add:
x
2
x
1 4
x
4
x
2 4 3
x
10 .
Solution x
2
x
1 4
x
4
x
2 4 3
x
10
x
1 (
x
2)(
x
2) 4 (
x
2)(
x
5) (
x
x
1 2)(
x
2)
x
5
x
5 4 (
x
2)(
x
5)
x
2
x
2 (
x x
2 2)(
x
6
x
5 2)(
x
5) 4
x
8 (
x
2)(
x
2)(
x
5)
x
2 6
x x
8 (
x
2)(
x
2)(
x
5) (
x x
2 2)(
x
10
x
3 2)(
x
5) Copyright © 2012 Pearson Education, Inc. Adding numerators
Slide 6- 21