<s>[INST] If $f(x)$ Polynomial with real coefficient and $f(0)=1$, $f(2)+f(3)=125$ and $f(x)*f(2x^2)=f(2x^3+x)$ then what is the value of $f(5)$? If $f(x)$ Polynomial with real coefficient and $f(0)=1$, $f(2)+f(3)=125$
and
$f(x)*f(2x^2)=f(2x^3+x)$
then what is the value of $f(5)$?
.
What I tried
$$f(0)=1$$
put x=1 $$f(1)*f(2)=f(3)$$
put x=2 $$f(2)*f(8)=f(18)$$
from this approach, I cant find f(5). Please Suggest a method to solve.
 [/INST] Here's a somewhat unfulfilling solution.
According to Solve $f(x)f(2x^2) = f(2x^3+x)$, the polynomial $f(x)$ is of the form $f(x)=(x^2+1)^n$ for some $n$.
Using $f(2)+f(3)=125$ we obtain $n=2$, so $f(x)=(x^2+1)^2$. Plugging in $x=5$ we conclude
$$f(5)=(5^2+1)^2=26^2=676.$$
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