Question

If $f\left(x\right)$ is a polynomial satisfying $f\left(x\right)\cdot f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ and $f\left(2\right)=17$, then the value of $f\left(3\right)$ is

• A. $33$
• B. $65$
• C. $82$
• D. $257$

Answer 1

The correct option is C $82$

Given that $f\left(x\right)\cdot f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$
$\Rightarrow \left(f\right(x)-1)(f\left(\frac{1}{x}\right)-1)=1$

Let $g\left(x\right)=f\left(x\right)-1$
$\Rightarrow g\left(x\right)\cdot g\left(\frac{1}{x}\right)=1\Rightarrow g\left(x\right)=\pm x^n\Rightarrow f\left(x\right)=1\pm x^n$

$f\left(2\right)=17\Rightarrow f\left(x\right)=1+x^4\therefore f\left(3\right)=82$