Definition of Rational Expressions
n
Rational Expressions
n
Peter Lo
Ma104 © Peter Lo 2002
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Basic Principle of Rational
Numbers
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A Rational Expressions is the ratio of two
polynomials with the denominator not equal to
zero.
The domain of a rational expressions is the set of
all real numbers that can be used in place of the
variable.
Ma104 © Peter Lo 2002
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Example
3
n
Reducing
− 2a 7b
a 2b 3
n
Reducing
2a 3 − 16
16 − 4a 2
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1
Reducing Rational Expressions
Example
n
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Multiplication and Division
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Mercedes Benz spent $700 million to develop its
new 1999 M class SUV, which will sell for around
$40,000. If the cost of manufacturing the SUV is
$30,000 each, then what rational function gives
the average cost of developing and manufacturing
x vehicles? Compare the average cost per vehicle
for manufacturing levels of 10,000 vehicles and
100,000 vehicles.
Ma104 © Peter Lo 2002
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Example
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n
Reducing
a3 − b3
6
⋅
b − a 2 a2 + 2 ab + 2b 2
n
Reducing
x2− 4
2
x −2
3
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2
Addition and Subtraction
Least Common Denominator
n
n
Example x + 4 x + 7
2
x2 − 1
−
To add fractions with denominators that are not identical,
Least Common Denominator (LCD) is the basic principle
to build up the denominators
x − 2 x +1
x2 − 1
2
Ma104 © Peter Lo 2002
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Fractions Operations
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Example
n
Simplify
1
2
+ 2
x −1 x + x
n
Simplify
5
3
−
a − 2 2− a
n
Simplify
2
x+2
2
x −9
x
4
+
x2 − 6x + 9 x − 3
Ma104 © Peter Lo 2002
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Ma104 © Peter Lo 2002
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3
Solving Equations involving
Rational Expressions
n
An Equation with Rational
Expressions
To solve an equation that contains rational
expressions:
u Find the LCD
u Multiply both sides of the equations by LCD
u Solve the resulting polynomial equation
u Check all solutions in the original equation.
Remove any extraneous solutions from the
solution set.
Ma104 © Peter Lo 2002
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14
An Equation with Extraneous
Root
An Equation with Two Solutions
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Ma104 © Peter Lo 2002
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Ma104 © Peter Lo 2002
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4
Proportions
n
A Proportion with One Solution
The equation ad = bc says that product of extremes is equal
to the product of the means. When solving a proportion,
we can omit multiplication by the LCD and just remember
the result, ad = bc, as the extremes-means property.
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A Proportion with Two Solutions
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Application Examples
n
Ma104 © Peter Lo 2002
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Michele drove her empty rig
300 miles to Salina to pick up
a load of cattle. When her rig
was fully loaded, her average
speed was 10 miles per hour
less than when the rig was
empty. If the return trip took
her 1 hour longer, then what
was her average speed with
the rig empty?
Ma104 © Peter Lo 2002
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