Question 143
Consider integers $b_1, b_2, \dots, b_n$ have the sum is $2022$ and take $a$ as a divisor of $\left\lfloor \dfrac{2022!}{3^{1006}} \right\rfloor$. Prove that the following polynomial does not have rational root
$$
P(x) = a x^{2n+2} + b_1 x^{2n} + b_2 x^{2n-2} + \dots + b_n x^2 + a.
$$
$$
P(x) = a x^{2n+2} + b_1 x^{2n} + b_2 x^{2n-2} + \dots + b_n x^2 + a.
$$