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Find a polynomial with n-th power variable

Source: Latvian TST for Baltic Way 2022 P4

November 24, 2022

algebrapolynomial

Problem Statement

Let

p(x)

be a polynomial with real coefficients and

n

be a positive integer. Prove that there exists a non-zero polynomial

q(x)

with real coefficients such that the polynomial

p(x)\cdot q(x)

has non-zero coefficients only by the powers which are multiples of

n