Q:

$f(x)$ is a polynomial satisfying $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2)$), given $f(2)=5.$

If f(x) is a polynomial satisfying  $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2))$, given $f(2)=5.$

ATTEMPT:- $f(f(2))=f(5)$, We can find $f(0)$,$f(1)$ and $f(1/2)$ to be $1,2$ and $5/4$ respectively.
we can change the function to the form $g(x)*g(y)=g(xy)$ by basic transformation.
but how to get $f(5)$ without using transformations.

A:

Hint: $(f(x)-1)(f(y)-1)=f(xy)-1$ so letting $h(x)=f(x)-1$ we have $h(x)h(y)=h(xy)$. Use that $h$ is a polynomial to show that $h(x)=x^n$ for some $n$.

