Transcript Chapter 6
Intermediate Algebra Chapter 6 - Gay •Rational Expressions Intermediate Algebra 6.1 • 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 6.1 •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. 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 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 6.2 • 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 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.” Intermediate Algebra 6.3 •Complex Fractions 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. Objective • Simplify a complex rational expression. 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.” Intermediate Algebra 6.4 •Division Long division Problems x 5x 7 x2 2 Long division Problems x 5x 7 x2 2 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 6.5 •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. Proportions and Cross Products • If a c where b, d 0 b d then ad bc Thomas Carlyle • “Ever noble work is at first impossible.” Intermediate Algebra 6.6 •Applications 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 Section 6.7 – Gay Variation and Problem Solving • • • • Direct Variation Inverse Variation Joint Variation Applications Def: Direct Variation • The value of y varies directly with the value of x if there is a constant k such that y = kx. Objective • Solve Direct Variation Problems • Determine constant of proportionality. 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. Example: Direct Variation • y varies directly as x. If y = 18 when x = 5, find y when x =8 • Answer: y = 28.8 Helen Keller – advocate for he blind • “Alone we can do so little, together we can do so much.” Definition: Inverse Variation • A quantity y varies inversely with x if there is a constant k such that k y x • y is inversely proportional to x. • k is called the constant of variation. 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. Joint Variation • Three variables y,x,z are in joint variation if y = kxz where k is a constant. Leonardo Da Vinci - scientist, inventor, and artist • “Time stays long enough for those who use it.”