TITLE: Find $f(5)$ of a non-constant polynomial function $f(x)$
QUESTION [7 upvotes]: Suppose $f(x)$ is a non-constant polynomial such that $f(x^ 3) − f(x ^ 3 − 2) = f( x )\cdot f(x) + 12$ for all $x$. Find $f(5)$?
I find this problem on Quora just now, and I try to solve it but do not know where to start (every time I substitue a number  $a$ into the equation, I get three more unknown numbers). Would anyone give me any clue? Is there a general method to deal with this kind of problems?

REPLY [4 votes]: Let $d$ be the degree of $f(x)$, $d > 0$. The degree of the RHS is $2d$. Now let's study the top degree terms of the LHS:
Write $f(x) = a x^d + b x^{d-1} + \dots$, $a\neq0$. then the LHS is (where $\dots$ are terms with degree $<3d-3$):
$$\begin{align}
ax^{3d} + b x^{3d-3} - \bigl( a (x^3-2)^d + b (x^3-2)^{d-1} \bigr) + \dots
& = ax^{3d} + bx^{3d-3} - \bigl( ax^{3d} - 2dax^{3d-3} + bx^{3d-3} \bigr) + \dots\\
&= -2dax^{3d-3} + \dots
\end{align}$$
So the LHS has degree 3d-3. Equating this with the degree of the RHS, the degree of $f(x)$ is 3. Now it's simply a matter of writing $f(x)$ explicitly, plugging into the original equation and solving for the coefficients.