Intermediate Algebra 098A Chapter 7 Rational Expressions.
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Transcript of Intermediate Algebra 098A Chapter 7 Rational Expressions.
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
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.
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.
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.
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.”
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 adding rational expressions with same
denominator
• 1. Add or subtract the numerators
• 2. Keep the same denominator.
• 3. Simplify to lowest terms.
Intermediate Algebra 098A 7.4
• Adding and Subtracting Rational Expressions with unlike Denominators
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
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.”
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.
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
Definition: Complex rational expression
• Is a rational expression that contains rational expressions in the numerator and denominator.
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.