Suppose-p-is-a-polynomial-with-complex-coefficients-and-an-even-degree-If-all-the-roots-of-p-are-complex-non-real-numbers-with-modulus-1-prove-that-p-1-R-iff-p-1-R-

Tinku Tara June 4, 2023 Algebra 0 Comments

Question Number 21309 by Tinkutara last updated on 20/Sep/17

Suppose p is a polynomial with complex coefficients and an even degree. If all the roots of p are complex non-real numbers with modulus 1, prove that p(1) ∈ R iff p(−1) ∈ R.

$$\mathrm{Suppose}\:{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{with}\:\mathrm{complex} \\ $$$$\mathrm{coefficients}\:\mathrm{and}\:\mathrm{an}\:\mathrm{even}\:\mathrm{degree}.\:\mathrm{If}\:\mathrm{all} \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{p}\:\mathrm{are}\:\mathrm{complex}\:\mathrm{non}-\mathrm{real} \\ $$$$\mathrm{numbers}\:\mathrm{with}\:\mathrm{modulus}\:\mathrm{1},\:\mathrm{prove}\:\mathrm{that} \\ $$$${p}\left(\mathrm{1}\right)\:\in\:{R}\:\mathrm{iff}\:{p}\left(−\mathrm{1}\right)\:\in\:{R}. \\ $$

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Suppose-p-is-a-polynomial-with-complex-coefficients-and-an-even-degree-If-all-the-roots-of-p-are-complex-non-real-numbers-with-modulus-1-prove-that-p-1-R-iff-p-1-R-

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