Let $f(x)$ be a polynomial such that $f(x)*f(1/x) +3f(x)+3f(1/x)=0$ and $f(3) =24$. What is $f(2) +f(-2)$? Let $f(x)$ be a polynomial such that $f(x)*f(1/x) +3f(x)+3f(1/x)=0$ and $f(3) =24$. What is $f(2) +f(-2)$?

Let $g(x)=f(x)+3$, then $g(x)g(\frac{1}{x})=9$ and $g(3)=27$.
Let $g(x)=ax^i+... +kx^j$ where $a$ and $k$ are the non-zero coefficients of the lowest and highest powers of $x$ in  $g(x)$.  Then
$$(ax^i+... +kx^j)(ax^{-i}+... +kx^{-j})=9$$
$$(ax^i+... +kx^j)(k+ ...+ax^{j-i})=9x^j$$
The LHS has a non-zero coefficient of $x^i$ and so we have $i=j$ i.e. $g(x)=ax^i$.  
Then $ax^i\times ax^{-i}=9$ and therefore $a=3$ or $-3$. Also, $a3^i=27$ and so $a=3,i=2$.
$f(x)=3x^2-3$ and $f(2)+f(-2)=18$.
