1. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §6.2 Rational Fcn Add & Subtract

2. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §6.1 → Rational Function Simplification  Any QUESTIONS About HomeWork §6.1 → HW-18 6.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 3 Bruce Mayer, PE Chabot College Mathematics Addition  The Sum of Two Rational Expressions  To add when the denominators are the same, add the numerators and keep the common denominator:

4. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 4 Bruce Mayer, PE Chabot College Mathematics Example  Rational Addition Add. Simplify the result, if possible. a) b) c) d)  SOLUTION a) The denominators are alike, so we add the numerators

5. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Rational Addition SOLUTION b)  SOLUTION c) The denominators are alike, so we add the numerators Combining like terms

6. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Rational Addition SOLUTION d) Factoring Combining like terms in the numerator ReMove a Multiplying “1”

7. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 7 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Rational Subtraction  Subtract. Simplify the result, if possible. a) b)  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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Rational Subtraction b) Removing the parentheses (using the distributive law) Factoring, in hopes of simplifying Removing a factor equal to 1

10. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 10 Bruce Mayer, PE Chabot College Mathematics Least Common: Multiples & Denominators  To add or subtract rational expressions that have different denominators, we must first find EQUIVALENT rational expressions that have a common denominator.  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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 11 Bruce Mayer, PE Chabot College Mathematics Find the Least Common Denom 1. Write the prime factorization of each denominator. 2. Select one of the factorizations and inspect it to see if it contains the other. a) If it does, it represents the LCM of the denominators. b) 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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  LCD  Find the LCD of: SOLUTION 1. Begin by writing the prime factorizations: 6x 2 = 2  3  x  x 4x 3 = 2  2  x  x  x 2. LCM = 2  2  3  x  x  x  The LCM of the denominators is thus 2 2  3  x 3, or 12x 3, so the LCD is 12x 3. Note that each factor appears the greatest number of times that it occurs in either of these factorizations.

13. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 13 Bruce Mayer, PE Chabot College Mathematics 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. The LCD requires additional factors of 2 and x The LCD requires another factor of 3.

14. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Least Common Mult  For each pair of polynomials, find the Least Common Multiple (LCM). a)16a and 24b b)24x 4 y 4 and 6x 6 y 2 c) x 2 – 4 and x 2 – 2x – 8

15. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 15 Bruce Mayer, PE Chabot College Mathematics 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 2 4  3  a  b, or 48ab 16a is a factor of the LCM 24b is a factor of the LCM

16. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  LCM  SOLUTION b) LCM for 24x 4 y 4 and 6x 6 y 2 24x 4 y 4 = 2  2  2  3  x  x  x  x  y  y  y  y 6x 6 y 2 = 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 24x 6 y 4

17. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  LCM  SOLUTION c) LCM for x 2 – 4 and x 2 – 2x – 8 x 2 – 4 = (x – 2)(x + 2) x 2 – 2x – 8 = (x + 2)(x – 4) LCM = (x – 2)(x + 2)(x – 4) x 2 – 4 is a factor of the LCM x 2 – 2x – 8 is a factor of the LCM

18. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Find equivalent expressions that have the LCD for This Expression Pair  SOLUTION From the previous example the LCD: (x  2)(x + 2)(x  4) Multiply by the missing expression

19. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Equiv. Expressions  We leave the results in factored form.  In a later slides we will carry out the actual addition and subtraction of such rational expressions. Multiply by the missing expression

20. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 20 Bruce Mayer, PE Chabot College Mathematics To Add or Subtract Rational Expressions Having Different Denominators 1.Find the LCD. 2.Multiply each rational expression by a Special form of 1 made up of the factors of the LCD missing from that expression’s denominator. 3.Add or subtract the numerators, as indicated. Write the sum or difference over the LCD. 4.Simplify, if possible

21. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 21 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Add 3.Next we add the numerators: 4.Since 16x 2 + 15x and 36 have no common factor, [16x 2 + 15x]/36 canNOT be simplified any further  Subtraction is performed in a very similar Fashion

23. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Subtract  SOLUTION: We follow the four steps as shown in the previous example. First, we find the LCD 9x = 3  3  x 12x 2 = 2  2  3  x  x LCD = 2  2  3  3  x  x = 36x 2  The denominator 9x must be multiplied by 4x to obtain the LCD.  The denominator 12x 2 must be multiplied by 3 to obtain the LCD.

24. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Subtract  Next, Multiply to obtain the LCD and then subtract and, if possible, simplify Caution! Do not simplify these rational expressions or you will lose the LCD. This cannot be simplified, so we are done.

25. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Add  SOLUTION: First, we find the LCD: a 2 – 4 = (a – 2)(a + 2) a 2 – 2a = a(a – 2)  Multiply by a form of 1 to obtain the LCD in each expression: LCD = a(a – 2)(a + 2)

26. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Add  Continue the Reduction  3a 2 + 2a + 4 does not factor so we are done

27. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 27 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Continue Reduction Removing parentheses and subtracting every term. When subtracting a numerator with more than one term, parentheses are important. 5x + 16 does not factor so we are finished

29. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Add  SOLUTION: 1 st Factor Denoms  Next Put AddEnds over the LCD  Now can Start the Reduction

30. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Add  The Reduction Adding numerators x 2 + 10x – 3 does not factor so we are finished

31. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 31 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 32 Bruce Mayer, PE Chabot College Mathematics Example  Add  SOLUTION by Reduction Multiplying by 1 using −1/−1 The denominators are now the same.

33. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 33 Bruce Mayer, PE Chabot College Mathematics Example  Add  SOLUTION by Reduction −3 + x = x + ( − 3) = x − 3

34. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 34 Bruce Mayer, PE Chabot College Mathematics Example  Add  SOLUTION by Reduction

35. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  Add  Complete the Reduction

36. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 36 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §6.2 Exercise Set 82 (ppt), 28, 64, 84  Add Rational Expressions

37. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 37 Bruce Mayer, PE Chabot College Mathematics P6.2-82  Risking a Ticket  If Drive-Time is to 8hr, 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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 38 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 39 Bruce Mayer, PE Chabot College Mathematics All Done for Today A Different Kind of LCM  Landing Craft, Mechanized

40. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 40 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

41. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 41 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

42. BMayer@ChabotCollege.edu MTH55_Lec-29_Fa08_sec_6-1_Rational_Fcn_Mult-n-Div.ppt 42 Bruce Mayer, PE Chabot College Mathematics