1. Specialist Mathematics
Polynomials
Week 4

2. Factorizing zn - an And zn + an

3. Example 27 (Ex 3H)

4. Solution 27

5. Example 28 (Ex 3H)

6. Solution 28

7. Quadratic Iterations
Iterative procedure involves repetition of the same process over and over.
We have a starting value zo, we do an iteration to produce z1, on which we do a further iteration to produce z2, etc.
We will be performing iterations of the form z2 + c, where c is complex.
Notation z z2 + c means f(z) = z2 + c.

8. Example 29 (Ex 3J1)

9. Solution 29

10. Investigation 3 Page 113

11. Summary for c = 0

12. Summary for c = 0 Invariant points where z1 = z0
2 cycle when bounces between 2 points.
3 cycle when bounces between 3 points.
n cycle when bounces between n points.
Chaotic behaviour when it neither converges, nor diverges nor exhibits cyclic behaviour

13. Example 30 (Ex 3J2)

14. Solution 30

15. Example 31(Ex 3J2)

16. Solution 31

17. Example 32 (Ex 3J2)

18. Solution 32

19. Julia Set

20. Julia Set for

21. Investigation 4 Page 117

22. Example 33 (Ex 3J3)

23. Solution 33

24. Mandelbrot Set

25. Investigation 5 Page 118

27. Properties of a Mandelbrot Set
The greater the number of iterations the more defined the set becomes.
All the points in the main body converge to a unique point
Points in each lobe give rise to cyclic behaviour.
Point exhibiting chaotic behaviour lie one extremities of the set.
In a lobe more iterations are needed before they cycle if further from the centre

28. Example 34 (Ex 3J4)

29. Solution 34

30. Example 35 (Ex 3J4)

31. Solution 35

32. Example 36 (Ex 3J4)

33. Solution 36

34. This Week Polynomials Study Guide Week 4. Page 109 Ex 3H Q1 - 7
Page 113 Ex3J1 Q1,2
Page 115 Ex3J2 Q1-5
Page 118 Ex3J3 Q1
Page 120 Ex3J4 Q1,2
Page 121 Review Sets 3A-3F