1. 11.4 – Multiplying & Dividing Rational Functions

2. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4

3. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2

4. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4

5. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z

6. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x

7. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x

8. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z

9. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z

10. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z 2

11. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z 2

12. Ex. 1 Find each product. a. 10y 3 z 2. 12w 2 x 2 6wx 3 25y 2 z 4 10y 3 z 2 · 12w 2 x 2 6wx 3 · 25y 2 z 4 2 · 5 · y · y · y · z · z · 2 · 6 · w · w · x · x 3 · 2 · w · x · x · x · 5 · 5 · y · y · z · z · z · z 4wy 5xz 2 2

13. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8

14. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8 (x – 5)(x + 4)

15. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8 (x – 5)(x + 4) (x + 9)(x + 11)

16. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8 (x – 5)(x + 4). (x + 11)(x + 2) (x + 9)(x + 11)

17. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8 (x – 5)(x + 4). (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4)

18. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8 (x – 5)(x + 4) (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4) (x – 5)(x + 4)(x + 11)(x + 2) (x + 9)(x + 11)(x + 2)(x + 4).

19. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8 (x – 5)(x + 4) (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4) (x – 5)(x + 4)(x + 11)(x + 2) (x + 9)(x + 11)(x + 2)(x + 4).

20. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8 (x – 5)(x + 4) (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4) (x – 5)(x + 4)(x + 11)(x + 2) (x + 9)(x + 11)(x + 2)(x + 4).

21. b. x 2 – x – 20. x 2 + 13x + 22 x 2 + 20x + 99 x 2 + 6x + 8 (x – 5)(x + 4) (x + 11)(x + 2) (x + 9)(x + 11) (x + 2)(x + 4) (x – 5)(x + 4)(x + 11)(x + 2) (x + 9)(x + 11)(x + 2)(x + 4) x – 5 x + 9.

22. Ex. 2 Find the product. 2.54 centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard

23. Ex. 2 Find the product. 2.54 centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard (2.54 centimeters)(12 inches)(3 feet)

24. Ex. 2 Find the product. 2.54 centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard (2.54 centimeters)(12 inches)(3 feet) (1 inch)(1 foot)(1 yard)

25. Ex. 2 Find the product. 2.54 centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard (2.54 centimeters)(12 inches)(3 feet) (1 inch)(1 foot)(1 yard)

26. Ex. 2 Find the product. 2.54 centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard (2.54 centimeters)(12 inches)(3 feet) (1 inch)(1 foot)(1 yard) (2.54cm.)(12)(3) 1 yd.

27. Ex. 2 Find the product. 2.54 centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard (2.54 centimeters)(12 inches)(3 feet) (1 inch)(1 foot)(1 yard) (2.54cm.)(12)(3) = 91.44cm. 1 yd. 1 yd.

28. Ex. 2 Find the product. 2.54 centimeters. 12 inches. 3 feet 1 inch 1 foot 1 yard (2.54 centimeters)(12 inches)(3 feet) (1 inch)(1 foot)(1 yard) (2.54cm.)(12)(3) = 91.44cm. = 91.44cm/yd 1 yd. 1 yd.

29. Ex. 3 How many inches are in a mile?

30. Ex. 3 How many inches are in a mile? Note: 5280 feet = 1 mile

31. Ex. 3 How many inches are in a mile? Note: 5280 feet = 1 mile 5280 feet 1 mile

32. Ex. 3 How many inches are in a mile? Note: 5280 feet = 1 mile 5280 feet 1 mile

33. Ex. 3 How many inches are in a mile? Note: 5280 feet = 1 mile 5280 feet · = inches 1 mile

34. Ex. 3 How many inches are in a mile? Note: 5280 feet = 1 mile 5280 feet · 12 inches 1 mile 1 foot

35. Ex. 3 How many inches are in a mile? Note: 5280 feet = 1 mile 5280 feet · 12 inches 1 mile 1 foot

36. Ex. 3 How many inches are in a mile? Note: 5280 feet = 1 mile 5280 feet · 12 inches = 63,360 in/mi 1 mile 1 foot

37. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3

38. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3

39. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2

40. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n

41. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p

42. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p

43. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n

44. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n

45. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n

46. Ex. Find each quotient. a. 16n 4 ÷ 12n 2 9p 2 3p 3 16n 4. 3p 3 9p 2 12n 2 2 · 2 · 2 · 2 · n · n · n · n· 3 · p · p · p 3 · 3 · p · p · 2 · 2 · 3 · n · n 4n 2 p 9

47. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3

48. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3

49. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2

50. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2

51. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3)

52. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3) (x + 3)(x – 3)

53. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2)

54. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) (x + 4)(x – 3)(x + 3)

55. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

56. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

57. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2)

58. b. x 2 + x – 12 ÷ x + 2 x 2 – 9 x + 3 x 2 + x – 12. x + 3 x 2 – 9 x + 2 (x + 4)(x – 3). (x + 3) (x + 3)(x – 3) (x + 2) (x + 4)(x – 3)(x + 3) (x + 3)(x – 3)(x + 2) x + 4 x + 2

59. c. ( x 2 + 6x – 16) ÷ (x – 2)

60. x 2 + 6x – 16

61. c. ( x 2 + 6x – 16) ÷ (x – 2) x 2 + 6x – 16 x – 2

62. c. ( x 2 + 6x – 16) ÷ (x – 2) x 2 + 6x – 16 x – 2 (x – 2)(x + 8) x – 2

63. c. ( x 2 + 6x – 16) ÷ (x – 2) x 2 + 6x – 16 x – 2 (x – 2)(x + 8) x – 2

64. c. ( x 2 + 6x – 16) ÷ (x – 2) x 2 + 6x – 16 x – 2 (x – 2)(x + 8) x – 2 x + 8