Question 117
Let polynomials $P(x)$ and $Q(x)$ have the same degree. Let $P_Q(x)$ denote the polynomial such that its coefficients of even degree variables coincide with the corresponding coefficients of $P(x)$ and its coefficients of odd degree variables coincide with the corresponding coefficients of $Q(x)$. (For example, if $P(x) = x^3 + 2x^2 + 4x + 1$ and $Q(x) = 3x^3 + x^2 + 2$, then $P_Q(x) = 3x^3 + 2x^2 + 1$, and $Q_P(x) = x^3 + x^2 + 4x + 2$.)
a) Prove that there exist $P(x)$ and $Q(x)$ such that they have no real roots but both $P_Q(x)$ and $Q_P(x)$ have at least one real root.
b) Find the smallest degree of $P(x)$ and $Q(x)$ satisfying a).
a) Prove that there exist $P(x)$ and $Q(x)$ such that they have no real roots but both $P_Q(x)$ and $Q_P(x)$ have at least one real root.
b) Find the smallest degree of $P(x)$ and $Q(x)$ satisfying a).