Question 178
Given three functions
$$
P(x) = (x^2 - 1)^{2023}, Q(x) = (2x + 1)^{14}, R(x) = \left(2x + 1 + \frac{2}{x}\right)^{34}.
$$
Initially, we pick a set $S$ containing two of these functions, and we perform some operations on it. Allowed operations include:
* Take two functions $p, q \in S$ and add one of $p+q, p-q$, or $pq$ to $S$.
* Take a function $p \in S$ and add $p^k$ to $S$ for $k$ is an arbitrary positive integer.
* Take a function $p \in S$ and choose a real number $t$, and add to $S$ one of the function $p+t, p-t, pt$.
Show that no matter how we pick $S$ in the beginning, there is no way we can perform finitely many operations on $S$ that would eventually yield the third function not in $S$.
$$
P(x) = (x^2 - 1)^{2023}, Q(x) = (2x + 1)^{14}, R(x) = \left(2x + 1 + \frac{2}{x}\right)^{34}.
$$
Initially, we pick a set $S$ containing two of these functions, and we perform some operations on it. Allowed operations include:
* Take two functions $p, q \in S$ and add one of $p+q, p-q$, or $pq$ to $S$.
* Take a function $p \in S$ and add $p^k$ to $S$ for $k$ is an arbitrary positive integer.
* Take a function $p \in S$ and choose a real number $t$, and add to $S$ one of the function $p+t, p-t, pt$.
Show that no matter how we pick $S$ in the beginning, there is no way we can perform finitely many operations on $S$ that would eventually yield the third function not in $S$.