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118Homework1

# 118Homework1 - Homework 1 Due Monday Aug 30 1 Find the cube...

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Homework 1 Due Monday, Aug. 30 1. Find the cube roots of 8 i . 2. If you consider the polynomial ax 2 + bx + c , the quadratic formula tells us that the roots of this polynomial are x = - b 2 a ± b 2 - 4 ac 2 a . Note that if a , b , and c are real, and if the polynomial has complex roots (i.e. b 2 - 4 ac < 0), then the roots are conjugates of each other. Prove that if z is a root of the polynomial a n x n + a n - 1 x n - 1 + . . . a 1 x + a 0 with a n , . . . , a 0 R , then z is also a root. (In other words, the complex roots of a polynomial with real coefficients occur in conjugate pairs.) 3. Let n be a positive integer greater than or equal to 2. (a) What is the sum of the n th roots of 1? (Prove that your claim is true.) (b) What is the product of the n th roots of 1? (Again, prove your claim.) 4. Suppose that z is a complex number lying on the circle of radius 2 centered at the origin (i.e., | z | = 2). Prove that z + 1 z 4 - 4 z 2 + 3 1 . (Note that the denominator factors.) 5. Prove that if z 1 and z 2 are complex numbers, then | z 1 | = | z 2 | if and only

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118Homework1 - Homework 1 Due Monday Aug 30 1 Find the cube...

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