1. Polynomial and Synthetic Division Objective: To solve polynomial equations by long division and synthetic division.

2. Long Division Let’s look at long division with integers.

3. Long Division Let’s look at long division with integers.

4. Long Division Let’s look at long division with integers.

5. Long Division Let’s look at long division with integers.

6. Long Division Let’s look at long division with integers.

7. Long Division Let’s look at long division with integers.

8. Long Division Divide by

9. Long Division Divide by What number times x makes it ?

10. Long Division Divide by What number times x makes it ?

11. Long Division Divide by What number times x makes it ?

12. Long Division Divide by What number times x makes it ?

13. Long Division You try:

14. Long Division You try:

15. Long Division You try:

16. Example 2 Divide by. Notice that I need to have every degree of x represented.

17. Example 2 Divide by.

18. Example 2 Divide by.

19. Example 2 Divide by.

20. Long Division In the last examples, we got a remainder of zero. This means that the divisor is a factor of the dividend. This will not always happen. We could have a remainder.

21. Example 3 Divide by

22. Example 3 Divide by

23. Example 3 Divide by

24. Example 3 Divide by So our answer is:

25. Example 3 You Try. Divide by

26. Example 3 Divide by So our answer is:

27. Synthetic Division There is another, easier way to divide polynomials. This is called synthetic division. We are only going to use the coefficients to find the answer.

28. Synthetic Division Divide by.

29. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 _________________

30. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 _________________ 5

31. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 _________________ 5

32. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 -15 _________________ 5 3

33. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 -15 _________________ 5 3

34. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 -15 -9 _________________ 5 3 -2

35. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 -15 -9 _________________ 5 3 -2

36. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 -15 -9 6 _________________ 5 3 -2 0

37. Synthetic Division Divide by. We are going to line up the coefficients and do a procedure that you need to memorize. -3| 5 18 7 -6 -15 -9 6 _________________ 5 3 -2 0 The answer is: with no remainder

38. Synthetic Division You try: Divide by. 2| 9 -18 -16 32 ____________________ 9

39. Synthetic Division You try: Divide by. 2| 9 -18 -16 32 18 0 -32 ____________________ 9 0 -16 0 The answer is:

40. Synthetic Division Divide by The difference here is that there is no, but every degree needs to be represented.

41. Synthetic Division Divide by -3| 1 0 -10 -2 4 ________________________ 1

42. Synthetic Division Divide by -3| 1 0 -10 -2 4 -3 9 3 -3 ________________________ 1 -3 -1 1 1 The answer is:

43. Synthetic Division You Try Divide by

44. Synthetic Division You Try Divide by 6| 3 -16 0 -72 18 12 72 __________________ 3 2 12 0 The answer is:

45. Synthetic Substitution Evaluate the function at x = -2.

46. Synthetic Substitution Evaluate the function at x = -2.

47. Synthetic Substitution Evaluate the function at x = -2. There is another way we can do this. We will do synthetic substitution again, but we will use -2 to divide.

48. Remainder Theorem Evaluate the function at x = -2. -2| 3 8 5 -7 -6 -4 -2 _________________ 3 2 1 -9 This means that f(-2) = -9.

49. Remainder Theorem You Try. Evaluate the function at x = 2.

50. Remainder Theorem You Try. Evaluate the function at x = 2. 2| 2 -3 7 -3 4 2 18 _________________ 2 1 9 15 This means that f(2) = 15.

51. Homework Page 295 5, 7, 9, 15, 19, 21, 25, 45