2.1

Part of 2019 Belarus Team Selection Test

Problems(1)

Existence problem about quadratic trinomial

Source: 2019 Belarus Team Selection Test 2.1

Given a quadratic trinomial

p(x)

with integer coefficients such that

p(x)

is not divisible by

3

for all integers

x

. Prove that there exist polynomials

f(x)

h(x)

with integer coefficients such that

p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1.

quadraticsalgebrapolynomial