Transcript Intermediate Algebra 098A

Intermediate Algebra 098A
Chapter 7
Rational Expressions
Intermediate Algebra 098A 7.1
•Introduction
•to
•Rational Expressions
Definition: Rational Expression
• Can be written as
P( x)
Q( x)
• Where P and Q are polynomials and Q(x) is
not 0.
Determine Domain of rational
function.
Solve the equation Q(x) = 0
• 2. Any solution of that
equation is a restricted value
and must be excluded from the
domain of the function.
• 1.
Graph
• Determine domain, range, intercepts
• Asymptotes
1
f ( x) 
x
Graph
• Determine domain, range, intercepts
• Asymptotes
1
g ( x)  2
x
Calculator Notes:
• [MODE][dot] useful
• Friendly window useful
• Asymptotes sometimes occur that are not
part of the graph.
• Be sure numerator and denominator are
enclosed in parentheses.
Fundamental Principle of
Rational Expressions
ac a

bc b
Simplifying Rational Expressions
to Lowest Terms
• 1. Write the numerator and
denominator in factored form.
• 2. Divide out all common
factors in the numerator and
denominator.
Negative sign rule
p p
p
 

q
q
q
Problem
y  4 (1)  y  4 

4  y  1 4  y 
1 y  4 


 1
y4
Objective:
• Simplify a Rational
Expression.
Denise Levertov – U. S. poet
• “Nothing is ever enough.
Images split the truth in
fractions.”
Robert H. Schuller
• “It takes but one positive
thought when given a chance
to survive and thrive to
overpower an entire army of
negative thoughts.”
Intermediate Algebra 098A 7.2
•Multiplication
•and
•Division
Multiplication of Rational
Expressions
• If a,b,c, and d represent algebraic
expressions, where b and d are not 0.
a c ac

b d bd
Procedure
• 1. Factor each numerator
and each denominator
completely.
• 2. Divide out common
factors.
Procedure
• 1. Factor each numerator
and each denominator
completely.
• 2. Divide out common
factors.
Procedure for Division
• Write down problem
• Invert and multiply
• Reduce
Objective:
•Multiply and divide
rational expressions.
John F. Kennedy – American
President
• “Don’t ask ‘why’, ask
instead, why not.”
Intermediate Algebra 098A 7.3
• Addition
•and
• Subtraction
Objective
• Add and Subtract
• rational expressions with
the same denominator.
Procedure adding rational
expressions with same
denominator
Add or subtract the
numerators
• 2. Keep the same denominator.
• 3. Simplify to lowest terms.
• 1.
Algebraic Definition
a b ab
 
c c
c
a b a b
 
c c
c
Intermediate Algebra 098A 7.4
• Adding and Subtracting Rational
Expressions with unlike Denominators
LCMLCD
• The LCM – least common
multiple of denominators is
called LCD – least common
denominator.
Objective
• Find the lest common denominator (LCD)
Determine LCM of polynomials
• 1. Factor each polynomial completely
– write the result in exponential form.
• 2. Include in the LCM each factor that
appears in at least one polynomial.
• 3. For each factor, use the largest
exponent that appears on that factor in
any polynomial.
Procedure: Add or subtract rational
expressions with different denominators.
• 1. Find the LCD and write down
• 2. “Build” each rational expression so
the LCD is the denominator.
• 3. Add or subtract the numerators and
keep the LCD as the denominator.
• 4. Simplify
Elementary Example
• LCD = 2 x 3
1 2 13 2 2
 


2 3 23 32
3 4 3 4 7
 

6 6
6
6
Objective
• Add and Subtract
• rational expressions with
unlike denominator.
Martin Luther
• “Even if I knew that
tomorrow the world
would go to pieces, I
would still plant my apple
tree.”
Maya Angelou - poet
• “Since time is the one
immaterial object which we
cannot influence – neither
speed up nor slow down, add
to nor diminish – it is an
imponderably valuable gift.”
Intermediate Algebra 098A 7.5
• Equations
•with
•Rational Expressions
Extraneous Solution
• An apparent solution that is a
restricted value.
Procedure to solve equations containing
rational expressions
• 1. Determine and write LCD
• 2. Eliminate the denominators of the
rational expressions by multiplying
both sides of the equation by the LCD.
• 3. Solve the resulting equation
• 4. Check all solutions in original
equation being careful of extraneous
solutions.
Graphical solution
• 1. Set = 0 , graph and look for x intercepts.
• Or
• 2. Graph left and right sides and look for
intersection of both graphs.
• Useful to check for extraneous solutions
and decimal approximations.
Thomas Carlyle
• “Ever noble work is at
first impossible.”
Intermediate Algebra 098A 7.6
• Applications
• Proportions and Problem
Solving
• With
• Rational Equations
Objective
• Use Problem Solving
methods including charts,
and table to solve problems
with two unknowns
involving rational
expressions.
Problems involving work
• (person’s rate of work) x
(person's time at work) =
amount of the task
completed by that person.
Work problems continued
• (amount completed by
one person) + (amount
completed by the other
person) = whole task
Intermediate Algebra 098A 7.7
• Simplifying Complex Fractions
Definition: Complex rational
expression
• Is a rational expression
that contains rational
expressions in the
numerator and
denominator.
Objective
• Simplify a complex
rational expression.
Procedure 1
• 1. Simplify the numerator and
denominator if needed.
• 2. Rewrite as a horizontal division
problem.
• 3. Invert and multiply
• Note – works best when fraction over
fraction.
Procedure 2
• 1. Multiply the numerator and
denominator of the complex rational
expression by the LCD of the
secondary denominators.
• 2. Simplify
• Note: Best with more complicated
expressions.
• Be careful using parentheses where
needed.
Paul J. Meyer
• “Enter every activity without
giving mental recognition to the
possibility of defeat.
Concentrate on your strengths,
instead of your weaknesses…on
your powers, instead of your
problems.”