4. Let P(z) be a polynomial (a) (3 points) Prove that (2) P(z) i 1...

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4. Let P(z) be a polynomial (a) (3 points) Prove that Р(2) P(z) Тт тi а1 2Z аn where a1 . .. , an are the roots of P and mi,.. . , mn E Z>o are their multiplicities. (Hint: what is the derivative of log P?) (b) (3 points) Suppose that c C is a positively oriented simple closed curve that avoids all the roots of P. Let N(P,T) be the number of roots of P that lie in the interior of I, counted with multiplicity. Prove that P'(z) dz Р() 1 N(P. Г). 2i7T 4. Let P(z) be a polynomial (a) (3 points) Prove that Р(2) P(z) Тт тi а1 2Z аn where a1 . .. , an are the roots of P and mi,.. . , mn E Z>o are their multiplicities. (Hint: what is the derivative of log P?) (b) (3 points) Suppose that c C is a positively oriented simple closed curve that avoids all the roots of P. Let N(P,T) be the number of roots of P that lie in the interior of I, counted with multiplicity. Prove that P'(z) dz Р() 1 N(P. Г). 2i7T