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find polynomials with real coefficients such zP(z) = 1 iff P(z-1)P(z + 1) = 0

Source: 3rd SAFEST Olympiad (July 2021) p5, South African - Estonian IMO teams

September 9, 2021

algebrapolynomialcomplex numbers

Problem Statement

Find all polynomials

P

with real coefficients having no repeated roots, such that for any complex number

z

, the equation

zP(z) = 1

holds if and only if

P(z-1)P(z + 1) = 0

.
Remark: Remember that the roots of a polynomial are not necessarily real numbers.