 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2022/9/a/i/solution

The right-hand side has modulus two and argument

π /3

. Its fifth roots therefore have

z_{k} = 2^{1/5} exp [i (\frac{π}{15} + \frac{2 πk}{5})], k = 0, 1, 2, 3, 4.

They are the vertices of a regular pentagon on the circle of radius

2^{1/5}

, beginning at angle

π /15

. These five roots are distinct, and a degree-five polynomial has no further complex roots.

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