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1. The problem statement, all variables and given/known data

Show that four of the five roots of z^{5}+ 15z + 1 = 0 belong to the annulus {z: 3/2 < |z| < 2}.

2. Relevant equations

Argument Principle (presumably)

3. The attempt at a solution

Since f(z) = z^{5}+ 15z + 1 is entire, it has no poles. Thus, if N_{1}is the number of zeros inside C_{2}(0), N_{2}is the number of zeroes inside C_{3/2}(0), and N is the number of zeroes inside the annulus {z: 3/2 < |z| < 2},

N = N_{2}- N_{1}= 1/(2πi) [ ∫_{C2(0)}f'(z)/f(z) dz - ∫_{C3/2(0)}f'(z)/f(z) dz ]

Right? Or is there an easier way?

EDIT: Ah, shoot! That assumes that |z|=3/2 and |z|=2 have no zeroes.

EDIT 2: And also that no zeroes are that |z|>2.

What's the strategy here?

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# Show that the four of the five roots of

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