Section 6.1 Rational Expression & Functions:
Definitions, Multiplying, Dividing
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
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Fractions - a Quick Review
Definitions:
Rational Functions, Expressions
Finding the Domains (and Exclusions) of
Rational Functions
Simplifying Rational Functions
Simplifying by factoring out -1
6.1
1
Fractions - Review

Q: When can you add or subtract fractions?
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Q: What do you do when denominators are not the same?

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A: Only when denominators are the same
A: Use their LCD to create equivalent fractions.
Q: How do you multiply fractions?

A: Factor all tops and factor all bottoms,
cancel matching factors, multiply tops and bottoms
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Q: What do you do first when dividing fractions?
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A: Turn division into multiplication : reciprocal the divisor.
Rational Expressions are Polynomial
Fractions ! Same rules! 2
6.1

Definitions
6.1
3
Finding the Domain (and exclusions)
of a Rational Function
Recall the domain of a function is the set of
all real numbers for which the function is defined.
-What real values make this function undefined
(divided by 0)?
Factor: x2 + 2x – 24 = (x – 4)(x + 6)
{x | x is Real, except for 4 or -6}
6.1
4
Graphs of Rational Functions
t=-5/2 is an Asymptote
2t + 5 ≠ 0
2t ≠ -5
t ≠ -5/2
6.1
5
Definitions
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Horizontal Asymptote – A horizontal line that
the graph of a function approaches as x values
get very large or very small.
Vertical Asymptote – A vertical line that the
graph of a function approaches as x values
approach a fixed number
6.1
6
More Properties of Fractions - Review
6.1
7
Simplifying Rational Expressions
(In general, the expressions are NOT equivalent)
 12 a b
4
 3 ab
2
4
x 9

2
x3
(  3 )( 4 ) aa b
3

(  3 ) ab b
2
2
2
( x  3 )( x  3 )
( x  3)
6.1

4a
b
3
2
 x3
8
First Factor and Identify domain exclusions,
Then Simplify
x  2 x  15
2
25  x
2
x  3x  9

( x  5 )( x  3 )
( 5  x )( 5  x )
x  27
2

( x  3 )( x  3 x  9 )
2 x  2 x  12
2
2
x  3 x  4 x  12
3
2
x   2, 3


1
x3
2 ( x  3 )( x  2 )
( x  3 )( x  2 )( x  2 )
6.1
x  5
(5  x )
( x  3 x  9)
2
3

( x  3)

x3
2
x2
9
Multiplying Fractions
a  6a  9
2
a

a
3
a3
( a  3) a
2

a ( a  3)
(First find domain exclusions)
Factor expressions,
then cancel like factors
6.1
3
 a ( a  3)
2
a  0, 3
10
Example – Step by Step
x≠0,3
x  6x  9
2
20 x

5x
2
x3
( x  6 x  9 )( 5 x )
2

( 20 x )( x  3 )
1 x
1
2

( x  3 )( x  3 )( 5 x )
2
( 20 x )( x  3 )
4 1
1.
2.
3.
4.
5.
6.
7.
Write down original problem
Combine with parentheses
Find any polynomials that need factoring
Rewrite (if any factoring was done)
Identify domain exclusions
Cancel out matching factors
6.1
Simplify the answer

x ( x  3)
4
1
11
Board Practice –
Rational Multiplication
3
a b
m
8
4

3a b
m
5
a  a  56 a  a  56

2
2
a  49
a  64
2
(2 x  x ) 
2
x
2
1.
2.
3.
4.
5.
6.
7.
x  xb  2 x  2 b
2
Write original problem
Combine w/ parens
Factor polynomials
Rewrite (if any factoring)
Identify domain exclusions
Cancel matching factors
6.1 Simplify the answer
12
Finding Powers of Rational Expressions
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Factor and Simplify (if possible) before applying the power
If part of a larger expression, see if any terms cancel out
Multiply out the terms in the numerator,
multiply out the terms in the denominator.
Leave in simplified factored form
2
2
2
 x5 
( x  5 )( x  5 )
( x  5)
 x5 
 
 2
 2
  
2
x
(
x

6
)
x
(
x

6
)
x
(
x

6
)
x
(
x

6
)
 x  6x 


6.1
13
Dividing Fractions
x 8
3
x 1
x  2x  4
2

3x  3x
2
Change Divide to
Multiply by Reciprocal,
follow multiply procedure
x 8
3

x 1
3x  3x
2

x  2x  4
2

 x  2  x  2 x  4  3 x  x  1 
 2
 3 xx  2 
 x  1
x  2 x  4 
2
6.1
14
Board Practice
- Rational Division
3
a b
m
1.
2.
3.
4.
5.
6.
7.
8
4

3a b
m
5
Write original problem
Combine w/ parens
Factor polynomials
Rewrite (if any factoring)
Identify domain exclusions
Cancel matching factors
Simplify the answer
x 8
3
4x  4
x 2 x  4
2

b  4b
2x  2
2
3
x 1
6.1
 (b  2 )
15
Mixed Operations
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Multiplications & Division are done left to right
In effect, make each divisor into a reciprocal
( x  x  6 )  ( x  3)  ( x  2 )  ( x  x  6 ) 
2
2
6.1
1

1
( x  3) ( x  2 )

( x  3 )( x  2 )
( x  3 )( x  2 )

x2
x2
16
What Next?

Present Section 6.2 Add/Subtract Rational Expressions
6.1
17