*Using Trig RepresentationRecall that a complex number can be
represented as
Then it follows that
What about z3 ?
*DeMoivre's TheoremIn general (a + bi)n is
Apply to
Try
*Using DeMoivre to Find RootsAgain, starting with a + bi =
also
works when n is a fractionThus we can take a root of a complex
number
*Using DeMoivre to Find RootsNote that there will be n such
roots
One each for k = 0, k = 1, k = n 1
Find the two square roots ofRepresent as z = r cis What is
r?What is ?
*Graphical Interpretation of RootsSolutions are:
Roots will be equally spaced around a circle with radius
r1/2
*Graphical Interpretation of RootsConsider cube root of 27
Using DeMoivre's Theorem
Roots will be equally spaced around a circle with radius
r1/3
*Roots of EquationsRecall that one method of solving polynomials
involves taking roots of both sidesx4 + 16 = 0x4 = - 64Now we can
determine the roots(they are all complex)
