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Bosnia and Herzegovina TST 1999 Day 2 Problem 3

Source: Bosnia and Herzegovina Team Selection Test 1999

September 20, 2018
algebrapolynomialrootsImaginary

Problem Statement

It is given polynomial P(x)=x4+3x3+3x+p,(p∈R)P(x)=x^4+3x^3+3x+p, (p \in \mathbb{R}) a)a) Find pp such that there exists polynomial with imaginary root x1x_1 such that ∣x1∣=1\mid x_1 \mid =1 and 2Re(x1)=12(17−3)2Re(x_1)=\frac{1}{2}\left(\sqrt{17}-3\right) b)b) Find all other roots of polynomial PP c)c) Prove that does not exist positive integer nn such that x1n=1x_1^n=1
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