Two polynomials with real coefficients have the leading coefficients equal to 1 . Each polynomial has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polynomial at the roots of the first one.
Sergey Yanzhinov
gnoka wrote:
2. Two polynomials with real coefficients have the leading coefficients equal to 1 . Each polynomial has an odd degree that is equal to the number of its distinct real roots. The product of the values of the first polynomial at the roots of the second polynomial is equal to 2024. Find the product of the values of the second polynomial at the roots of the first one.
Let $P(x)=\prod_{i=1}^n(x-p_i)$ and $Q(x)=\prod_{j=1}^m(x-q_i)$
$2024=\prod_{j=1}^mP(q_j)=\prod_{j=1}^m\prod_{i=1}^n(q_j-p_i)$
And $\prod_{i=1}^nQ(p_i)=\prod_{i=1}^n\prod_{j=1}^m(p_i-q_j)=(-1)^{mn}2024=\boxed{-2024}$