Intermediate Algebra Chapter 6 - Gay Rational Expressions.
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Transcript of Intermediate Algebra Chapter 6 - Gay Rational Expressions.
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Intermediate Algebra Chapter 6 - Gay
•Rational Expressions
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Intermediate Algebra 6.1
•Introduction
• to
•Rational Expressions
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Definition: Rational Expression
• Can be written as
• Where P and Q are polynomials and Q(x) is not 0.
Determine domain, range, intercepts
( )
( )
P x
Q x
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Determine Domain of rational function.
• 1. 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.
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Graph
• Determine domain, range, intercepts
• Asymptotes
1( )f x
x
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Graph
• Determine domain, range, intercepts
• Asymptotes
2
1( )g x
x
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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.
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Fundamental Principle of Rational Expressions
ac a
bc b
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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.
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Negative sign rule
p p p
q q q
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Problem
( 1) 44
4 1 4
1 41
4
yy
y y
y
y
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Objective:
•Simplify a Rational Expression.
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Denise Levertov – U. S. poet
• “Nothing is ever enough. Images split the truth in fractions.”
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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.”
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Intermediate Algebra 6.1
•Multiplication
•and
•Division
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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
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Procedure
• 1. Factor each numerator and each denominator completely.
• 2. Divide out common factors.
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Definition of Division of Rational Expressions
• If a,b,c,and d represent algebraic expressions, where b,c,and d are not 0
a c a d ad
b d b c bc
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Procedure for Division
• Write down problem
• Invert and multiply
• Reduce
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Objective:
•Multiply and divide rational expressions.
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John F. Kennedy – American President
•“Don’t ask ‘why’, ask instead, why not.”
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Intermediate Algebra 6.2
•Addition
•and
•Subtraction
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Objective
• Add and Subtract • rational expressions with
the same denominator.
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Procedure adding rational expressions with same
denominator
• 1. Add or subtract the numerators
• 2. Keep the same denominator.
• 3. Simplify to lowest terms.
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Algebraic Definition
a b a b
c c ca b a b
c c c
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LCMLCD
• The LCM – least common multiple of denominators is called LCD – least common denominator.
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Objective
• Find the lest common denominator (LCD)
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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.
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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
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Elementary Example
• LCD = 2 x 3
1 2 1 3 2 2
2 3 2 3 3 2
3 4 3 4 7
6 6 6 6
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Objective
• Add and Subtract • rational expressions with
unlike denominator.
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Martin Luther
• “Even if I knew that tomorrow the world would go to pieces, I would still plant my apple tree.”
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Intermediate Algebra 6.3
•Complex Fractions
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Definition: Complex rational expression
• Is a rational expression that contains rational expressions in the numerator and denominator.
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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.
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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.
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Objective
• Simplify a complex rational expression.
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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.”
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Intermediate Algebra 6.4
•Division
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Long division Problems
2 5 7
2
x x
x
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Long division Problems
2 5 7
2
x x
x
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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.”
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Intermediate Algebra 6.5
•Equations
•with
•Rational Expressions
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Extraneous Solution
• An apparent solution that is a restricted value.
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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.
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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.
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Proportions and Cross Products
• If
, 0a c
where b db dthen ad bc
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Thomas Carlyle
•“Ever noble work is at first impossible.”
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Intermediate Algebra 6.6
•Applications
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Objective
• Use Problem Solving methods including charts, and table to solve problems with two unknowns involving rational expressions.
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Problems involving work
• (person’s rate of work) x (person's time at work) = amount of the task completed by that person.
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Work problems continued
• (amount completed by one person) + (amount completed by the other person) = whole task
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Section 6.7 – GayVariation and Problem Solving
• Direct Variation
• Inverse Variation
• Joint Variation
• Applications
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Def: Direct Variation
• The value of y varies directly with the value of x if there is a constant k such that y = kx.
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Objective
• Solve Direct Variation Problems
• Determine constant of proportionality.
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Procedure:Solving Variation Problems
• 1. Write the equation • Example y = kx• 2. Substitute the initial values and
find k.• 3. Substitute for k in the original
equation• 4. Solve for unknown using new
equation.
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Example: Direct Variation
• y varies directly as x. If y = 18 when x = 5, find y when x = 8
• Answer: y = 28.8
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Helen Keller – advocate for he blind
•“Alone we can do so little, together we can do so much.”
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Definition: Inverse Variation
• A quantity y varies inversely with x if there is a constant k such that
• y is inversely proportional to x.
• k is called the constant of variation.
ky
x
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Procedure: Solving inverse variation problems
• 1. Write the equation• 2. Substitute the initial values
and find k• 3. Substitute for k in the
equation found in step 1.• 4. Solve for the unknown.
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Joint Variation
• Three variables y,x,z are in joint variation if y = kxz where k is a constant.
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Leonardo Da Vinci - scientist, inventor, and artist
•“Time stays long enough for those who use it.”