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DeMoivre's Theorem Lesson 5.3
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2 Using Trig Representation Recall that a complex number can be represented as Then it follows that What about z 3 ?
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3 DeMoivre's Theorem In general (a + bi) n is Apply to Try
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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
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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?
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6 Graphical Interpretation of Roots Solutions are: Roots will be equally spaced around a circle with radius r 1/2
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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
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8 Roots of Equations Recall that one method of solving polynomials involves taking roots of both sides x 4 + 16 = 0 x 4 = - 64 Now we can determine the roots (they are all complex) Try out spreadsheet for complex roots
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9 Assignment Lesson 5.3 Page 354 Exercises 1 – 41 EOO
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