MathDB

For a non-constant polynomial

P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} \in \mathbb{R}[x], a_{n} \neq 0, n \in \mathbb{N}

, we say that

P

is symmetric if

a_{k}=a_{n-k}

for every

k=0,1, \ldots,\left\lceil\frac{n}{2}\right\rceil

. We define the weight of a non-constant polynomial

P \in \mathbb{R}[x]

, denoted by

t(P)

, as the multiplicity of its zero with the highest multiplicity.
a) Prove that there exist non-constant, monic, pairwise distinct polynomials

P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]

, none of which is symmetric, such that the product of any two (distinct) polynomials is symmetric.
b) What is the smallest possible value of

t\left(P_{1} \cdot P_{2} \cdot \ldots \cdot P_{2021}\right)

, if

P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]

are non-constant, monic, pairwise distinct polynomials, none of which is symmetric, and the product of any two (distinct) polynomials is symmetric?

P1