Specialist Mathematics Polynomials Week 4

Factorizing zn - an And zn + an

Example 27 (Ex 3H)

Solution 27

Example 28 (Ex 3H)

Solution 28

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.

Example 29 (Ex 3J1)

Solution 29

Investigation 3 Page 113

Summary for c = 0

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

Example 30 (Ex 3J2)

Solution 30

Example 31(Ex 3J2)

Solution 31

Example 32 (Ex 3J2)

Solution 32

Julia Set

Julia Set for .

Investigation 4 Page 117

Example 33 (Ex 3J3)

Solution 33

Mandelbrot Set

Investigation 5 Page 118

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

Example 34 (Ex 3J4)

Solution 34

Example 35 (Ex 3J4)

Solution 35

Example 36 (Ex 3J4)

Solution 36

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