MathDB

A polynomial

Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0

with real coefficients is called powerful if the equality

| k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n |

, and non-increasing , if

k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n

. Let for the polynomial

P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0

with nonzero real coefficients, where

a_d> 0

, the polynomial

P (x) (x-1) ^ t (x + 1) ^ s

is powerful for some non-negative integers

s

and

t

(

s + t> 0

). Prove that at least one of the polynomials

P (x)

and

(- 1) ^ d P (-x)

is nonincreasing.

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