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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 y4 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 ab c c c a b a b c c c Intermediate Algebra 098A 7.4 • Adding and Subtracting Rational Expressions with unlike Denominators LCMLCD • 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.”