Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §6.2 Rational FcnAdd & Subtract Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. MTH 55 6.1 Review § • Any QUESTIONS About • §6.1 → Rational Function Simplification • Any QUESTIONS About HomeWork • §6.1 → HW-23

3. Addition • The Sum of Two Rational Expressions • To add when the denominators are the same, add the numerators and keep the common denominator:

4. Example  Rational Addition Add. Simplify the result, if possible. b) a) d) c) • SOLUTION a) The denominators are alike, so we add the numerators

5. Example  Rational Addition SOLUTION b) The denominators are alike, so we add the numerators • SOLUTION c) Combining like terms

6. Example  Rational Addition SOLUTION d) Combining like terms in the numerator Factoring ReMove a Multiplying “1”

7. Subtraction • The Difference of Two Rational Expressions • To subtract when the denominators are the same, subtract the second numerator from the first and keep the common denominator:

8. Example  Rational Subtraction • Subtract. Simplify the result, if possible. b) a) • SOLUTION a) The parentheses are needed to make sure that we subtract both terms. Removing the parentheses and changing the signs (using the distributive law) Combining like terms

9. Example  Rational Subtraction b) Removing the parentheses (using the distributive law) Factoring, in hopes of simplifying Removing a factor equal to 1

10. Least Common: Multiples & Denominators • To add or subtract rational expressions that have different denominators, we must first find EQUIVALENT rational expressions that have a commondenominator. • The least common denom must include the factors of each number, so it must include each prime factor the greatest number of times that it appears in any of the factorizations of any denom.

11. Find the Least Common Denom • Write the prime factorization of each denominator. • Selectone of the factorizations and inspect it to see if it contains the other. • If it does, it represents the LCM of the denominators. • If it does not, multiply that factorization by any factors of the other denominator that it lacks. The final product is the LCM of the denominators. • The LCD is the LCM of the denominators. It should contain each factor the greatest number of times that it occurs in any of the individual factorizations.

12. Example  LCD • Find the LCD of: SOLUTION 1. Begin by writing the prime factorizations: 6x2 = 2  3  x  x 4x3 = 2  2  x  x  x 2. LCM = 2  2  3  x  x  x • The LCM of the denominators is thus 22  3  x3, or 12x3, so the LCD is 12x3. Note that each factor appears the greatest number of times that it occurs in either of these factorizations.

13. The LCD requires another factor of 3. The LCD requires additional factors of 2 and x Example  LCD cont. • Now Can Add • To obtain equivalent expressions with the LCD, we must multiply each expression by 1, using the missing factors of the LCD to write the 1.

14. Example  Least Common Mult • For each pair of polynomials, find the Least Common Multiple (LCM). • 16a and 24b • 24x4y4 and 6x6y2 • x2– 4 and x2– 2x– 8

15. Example  LCM • SOLUTION a) 16a = 2  2  2  2  a 24b = 2  2  2  3  b The LCM =2  2  2  2  a  3  b The LCM is 24  3  a  b, or 48ab 16a is a factorof the LCM 24b is a factor of the LCM

16. Example  LCM • SOLUTION b) LCM for 24x4y4 and 6x6y2 24x4y4 = 2  2  2  3  x  x  x  x  y  y  y  y 6x6y2 = 2  3  x  x  x  x  x  x  y  y LCM = 2  2  2  3  x  x  x  x y  y  y  y x  x • Note that we used the highest power of each factor. The LCM is 24x6y4

17. Example  LCM • SOLUTION c) LCM for x2– 4 and x2– 2x– 8 x2– 4 is a factor of the LCM x2– 4 = (x– 2)(x + 2) x2– 2x– 8 = (x + 2)(x– 4) LCM = (x– 2)(x + 2)(x– 4) x2– 2x – 8 is a factor of the LCM

18. Example • Find equivalent expressionsthat have the LCD for This Expression Pair • SOLUTIONFrom the previous example the LCD:(x  2)(x + 2)(x  4) Multiply by the missing expression

19. Example  Equiv. Expressions Multiply by the missing expression • We leave the results in factored form. • In a later slides we will carry out the actual addition and subtraction of such rational expressions.

20. To Add or Subtract Rational Expressions Having Different Denominators • Find the LCD. • Multiply each rational expression by a Special form of 1 made up of the factors of the LCD missing from that expression’s denominator. • Add or subtract the numerators, as indicated. Write the sum or difference over the LCD. • Simplify, if possible

21. Example  Add • SOLUTION 1. First, find the LCD: 9 = 3  3 12 = 2  2  3 2. Multiply each expression by the appropriate “form of 1” to get the LCD. LCD = 2  2  3  3 = 36

22. Example  Add Next we add the numerators: • Since 16x2 + 15x and 36 have no common factor, [16x2 + 15x]/36 canNOT be simplified any further • Subtraction is performed in a very similar Fashion

23. Example  Subtract • SOLUTION: We follow the four steps as shown in the previous example. First, we find the LCD 9x = 3  3  x 12x2 = 2  2  3  x  x LCD = 2  2  3  3  x  x = 36x2 • The denominator 9x must be multiplied by 4x to obtain the LCD. • The denominator 12x2 must be multiplied by 3 to obtain the LCD.

24. Caution!Do not simplify these rational expressions or you will lose the LCD. Example  Subtract • Next, Multiply to obtain the LCD and then subtract and, if possible, simplify This cannot be simplified, so we are done.

25. Example  Add • SOLUTION: First, we find the LCD: a2– 4 = (a– 2)(a + 2) a2– 2a = a(a– 2) LCD = a(a– 2)(a + 2) • Multiply by a form of 1 to obtain the LCD in each expression:

26. Example  Add • Continue the Reduction • 3a2 + 2a + 4 does not factor so we are done

27. Example  Subtract • SOLUTION: First, we find the LCD. It is just the product of the denominators: LCD = (x + 4)(x + 6). • We multiply by a form of 1 to get the LCD in each expression. Then we subtract and try to simplify Multiplying out numerators.

28. When subtracting a numerator with more than one term, parentheses are important. Example  Continue Reduction Removing parentheses and subtracting every term. 5x + 16 does not factor so we are finished

29. Example  Add • SOLUTION: 1st Factor Denoms • Next Put AddEnds over the LCD • Now can Start the Reduction

30. Example  Add • The Reduction Adding numerators x2 + 10x– 3 does not factor so we are finished

31. When Factors are Opposites • When one denominator is the opposite of the other, we can first multiply either expression by 1 using –1/ –1.

32. Example  Add • SOLUTION by Reduction Multiplying by 1 using −1/−1 The denominators are now the same.

33. Example  Add • SOLUTION by Reduction −3 + x = x + (−3) = x− 3

34. Example  Add • SOLUTION by Reduction

35. Example  Add • Complete the Reduction

36. WhiteBoard Work • Problems From §6.2 Exercise Set • 82 (ppt), 28, 64, 84 • Add Rational Expressions

37. P6.2-82  Risking a Ticket • If Drive-Time is to be 8hrs, how much over the 70mph & 65mph Spd Limits is required • ID 8hrs on Graph and find the OverSpeed needed to make this time • ANS → need to go about 22 mph over the Speed Limit for the entire 8 hrs

38. P6.2-82  Risking a Ticket • Is 8hrs too fast for this trip? • Find the average speed = Dist/Time • Total Distance = 470mi + 250mi = 720mi • Avg Speed = [720mi]/[8hrs] = 90 miles/hr • WOW! Running at 90 mph for 8 straight hours is NOT a realistic travel Plan • Better to try 10 hrs for an avg speed of 72 mph

39. All Done for Today • Landing Craft, Mechanized A DifferentKind of LCM

40. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

41. Graph y = |x| • Make T-table