DeMoivre's Theorem Lesson 5.3

2 Using Trig Representation  Recall that a complex number can be represented as  Then it follows that  What about z 3 ?

3 DeMoivre's Theorem  In general (a + bi) n is  Apply to  Try

4 Using DeMoivre to Find Roots  Again, starting with a + bi =  also works when n is a fraction Thus we can take a root of a complex number

5 Using DeMoivre to Find Roots  Note that there will be n such roots One each for k = 0, k = 1, … k = n – 1  Find the two square roots of Represent as z = r cis θ What is r? What is θ?

6 Graphical Interpretation of Roots  Solutions are: Roots will be equally spaced around a circle with radius r 1/2

7 Graphical Interpretation of Roots  Consider cube root of 27  Using DeMoivre's Theorem Roots will be equally spaced around a circle with radius r 1/3

8 Roots of Equations  Recall that one method of solving polynomials involves taking roots of both sides x = 0 x 4 = - 64  Now we can determine the roots (they are all complex) Try out spreadsheet for complex roots

9 Assignment  Lesson 5.3  Page 354  Exercises 1 – 41 EOO