DeMoivre's Theorem

1. 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 z3 ?

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 r1/2

7. • Graphical Interpretation of Roots • Consider cube root of 27 • Using DeMoivre's Theorem Roots will be equally spaced around a circle with radius r1/3

8. Roots of Equations • Recall that one method of solving polynomials involves taking roots of both sides • x4 + 16 = 0x4 = - 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