Intermediate Algebra 098A

1. Intermediate Algebra 098A Chapter 7 Rational Expressions

2. Intermediate Algebra 098A 7.1 • Introduction • to • Rational Expressions

3. Definition: Rational Expression • Can be written as • Where P and Q are polynomials and Q(x) is not 0.

4. 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.

5. Graph • Determine domain, range, intercepts • Asymptotes

6. Graph • Determine domain, range, intercepts • Asymptotes

7. 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.

8. Fundamental Principle of Rational Expressions

9. 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.

10. Negative sign rule

11. Problem

12. Objective: • Simplify a Rational Expression.

13. Denise Levertov – U. S. poet • “Nothing is ever enough. Images split the truth in fractions.”

14. 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.”

15. Intermediate Algebra 098A 7.2 • Multiplication • and • Division

16. Multiplication of Rational Expressions • If a,b,c, and d represent algebraic expressions, where b and d are not 0.

17. Procedure • 1. Factor each numerator and each denominator completely. • 2. Divide out common factors.

18. Procedure • 1. Factor each numerator and each denominator completely. • 2. Divide out common factors.

19. Procedure for Division • Write down problem • Invert and multiply • Reduce

20. Objective: • Multiply and divide rational expressions.

21. John F. Kennedy – American President • “Don’t ask ‘why’, ask instead, why not.”

22. Intermediate Algebra 098A 7.3 • Addition • and • Subtraction

23. Objective • Add and Subtract • rational expressions with the same denominator.

24. Procedure adding rational expressions with same denominator • 1. Add or subtract the numerators • 2. Keep the same denominator. • 3. Simplify to lowest terms.

25. Algebraic Definition

26. Intermediate Algebra 098A 7.4 • Adding and Subtracting Rational Expressions with unlike Denominators

27. LCMLCD • The LCM – least common multiple of denominators is called LCD – least common denominator.

28. Objective • Find the lest common denominator (LCD)

29. 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.

30. 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

31. Elementary Example • LCD = 2 x 3

32. Objective • Add and Subtract • rational expressions with unlike denominator.

33. Martin Luther • “Even if I knew that tomorrow the world would go to pieces, I would still plant my apple tree.”

34. 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.”

35. Intermediate Algebra 098A 7.5 • Equations • with • Rational Expressions

36. Extraneous Solution • An apparent solution that is a restricted value.

37. 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.

38. 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.

39. Thomas Carlyle • “Ever noble work is at first impossible.”

40. Intermediate Algebra 098A 7.6 • Applications • Proportions and Problem Solving • With • Rational Equations

41. Objective • Use Problem Solving methods including charts, and table to solve problems with two unknowns involving rational expressions.

42. Problems involving work • (person’s rate of work) x (person's time at work) = amount of the task completed by that person.

43. Work problems continued • (amount completed by one person) + (amount completed by the other person) = whole task

44. Intermediate Algebra 098A 7.7 • Simplifying Complex Fractions

45. Definition: Complex rational expression • Is a rational expression that contains rational expressions in the numerator and denominator.

46. Objective • Simplify a complex rational expression.

47. 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.

48. 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.

49. 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.”