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Copyright © 2007 Pearson Education, Inc. Slide R-1
Rational Expressions and Equations
Chapter 9
pg 470
Copyright © 2007 Pearson Education, Inc. Slide R-2
Review of Rational Expressions
• A rational expression is an expression that is the quotient of two polynomials.
Examples include
2
2
6 ( 6)( 4) 2 7 4, ,
2 ( 2)( 4) 5 20
x x x p p
x x x p p
Copyright © 2007 Pearson Education, Inc. Slide R-3
Domain of a Rational Expression
• The domain of a rational expression is the set of real numbers for which the expression is defined.
• The domain consists of all real numbers except those that make the denominator 0.
Copyright © 2007 Pearson Education, Inc. Slide R-4
Domain of a Rational Expression
For example, to find the domain of
solve as follows,
or
or
The domain is
( 6)( 4)
( 2)( 4)
x x
x x
( 2)( 4) 0x x
2 0x 4 0x
2x 4x
{ | 2, 4} .x x
Copyright © 2007 Pearson Education, Inc. Slide R-5
Lowest Terms of a Rational Expression
Fundamental Principle of Fractions
( 0 , 0)ac a
b cbc b
Copyright © 2007 Pearson Education, Inc. Slide R-6
Writing Rational Expressions in Lowest Terms
Example Write each rational expression in lowest terms.
(a) (b)
Solution
(a)
by the fundamental principle, provided p is not 0 or –4.
2
2
2 7 4
5 20
p p
p p
2
2
2 7 4 (2 1) 2 1( 4)
5 20 5 5( 4)
p p p p
p p p p
p
p
2
6 3
4
k
k
Copyright © 2007 Pearson Education, Inc. Slide R-7
Writing Rational Expressions in Lowest Terms
Solution
(b)
by the fundamental principle.
2
6 3 3 3
4 ( 2) ( 2
(2 ) (2 )( 1)
( 2) ( 2)( 1)
(
)
3 3
( 2)
2 )( 1)
2) ( )(2
k k
k k
k
k
k
k k k
k k
Copyright © 2007 Pearson Education, Inc. Slide R-8
Multiplying and Dividing Rational Expressions
Multiplying and Dividing Fractions
For fractions and
and
( 0 , 0),c
b dd
a
b
a c ac
b d bd , if 0.
a c a d c
b d b c d
Copyright © 2007 Pearson Education, Inc. Slide R-9
Multiplying and Dividing Rational Expressions
Example Multiply or divide as indicated.
(a) (b)
Solution
(a)
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
2
5
2 27
9 8
y
y
2
5 3
3
2
2
2
5
22
2
27 2 27 3
9 8 9 8
3
4
9 4
9 y
y
y y
y y y
y
Copyright © 2007 Pearson Education, Inc. Slide R-10
Multiplying and Dividing Rational Expressions
Solution (b)
2
3 2 4 3 2 3
3
2
3
2
3 11 4 9 36 ( 4)(3 1) 9( 4)
24 8 24 36 8 (3 1) 12 (2 3)
(12 )(2 3)
8 (9)
12 (2 3)
9 8
(2 3)
( 4)
(
(3 1)
(3 1 ))
6
4
p p p p p p
p p p p p p p p
pp
p
p
p
p
p
p p
p
p p
Copyright © 2007 Pearson Education, Inc. Slide R-11
Complex Fractions
• Complex fractions are those fractions whose numerator & denominator both contain fractions.
Copyright © 2007 Pearson Education, Inc. Slide R-12
Now you try!
• Pg 476- 477
• #’s 14-22 evens
• #’s 26-34 evens
• #’s 36, 37, 38
Copyright © 2007 Pearson Education, Inc. Slide R-13
Adding and Subtracting Rational Expressions
Adding and Subtracting Fractions
For fractions and
and
( 0 , 0),c
b dd
a
b
a c ad bc
b d bd
.
a c ad bc
b d bd
•Addition and subtraction are typically performed using the least common denominator.
Copyright © 2007 Pearson Education, Inc. Slide R-14
Adding and Subtracting Rational Expressions
Finding the Least Common Denominator (LCD)
1. Write each denominator as a product of prime factors.
2. Form a product of all the different prime factors. Each factor should have as exponent the greatest exponent that appears on that factor.
Copyright © 2007 Pearson Education, Inc. Slide R-15
Adding/Subtracting
• when we talk about CDs, we mean denominators that contain the same factors.
• To find our CD, we will first factor the ones we have.
• Then we will multiply each denominator by the factors it is missing to create a CD.
• Remember, we must also multiply the numerator by that same factor.
Copyright © 2007 Pearson Education, Inc. Slide R-16
Adding and Subtracting Rational Expressions
Example Add or subtract, as indicated.
(a) (b)
Solution
(a) Step 1: Find the LCD
2 2
2 3
2 4 2
y y
y y y y
2
5 1
9 6x x
Copyright © 2007 Pearson Education, Inc. Slide R-17
Adding and Subtracting Rational Expressions
Solution (a) The LCD is
Then1 2 2 22 3 18 .x x
2 2
2 2
2
5 1 5 1
9 6 9 610 3
18 181
2
.
2
0
3
3
3
18
x
xx x x xx
x xx
x
Copyright © 2007 Pearson Education, Inc. Slide R-18
Adding and Subtracting Rational Expressions
Solution (b)
2 2 2
2
2 2 2 2
2 2 2
2
2
2 3 2 3
2 4 2 ( 1) 2( 1)
( 2) 3
( 1) 2( 1)
2( 2) 3 2 2 4 3
2 ( 1) 2 ( 1) 2
2( 1
( 1)
2 4
)
2(
2 (
)
1
1
)
y y y y
y y y y y y y
y y
y y y
y y y y y y
y y y y y y
y y
y
y
y
y
y y
Copyright © 2007 Pearson Education, Inc. Slide R-19
Now you try!
• Pg 482- 483 #’s 26-36 evens
Copyright © 2007 Pearson Education, Inc. Slide R-20
Complex Fractions
• A complex fraction is any quotient of two rational expressions.
Copyright © 2007 Pearson Education, Inc. Slide R-21
Simplifying Complex Fractions
Example Simplify
Solution
Multiply both numerator and denominator by the LCD of all the fractions a(a + 1).
11
1 11
aa a
a a
Copyright © 2007 Pearson Education, Inc. Slide R-22
Simplifying Complex Fractions
Solution
2 2
( 1) ( 1)11 1
11 11 1 1 11 1
1 1
( 1)
( 11
( 1) 1
( 1) 2
) 1
1
( )( 1)
a a a a aaa aa aa a a a
a a a aa a
a a a a
a a
a
a a
a
a aa a
Copyright © 2007 Pearson Education, Inc. Slide R-23
Now you try!
• Pg 483 #’s 40