Fibonacci Polynomial Coefficients & Constant!

Let $f(x) = x^n + a_{n-1}x^{n - 1} + \dots + a_0$ for integer $n \geq 1$ , where $a_n = a_{n + 1} + a_{n + 2}$ with $a_{n - 1} = 1, a_{n - 2} = 2$ . Which of the following is/are true:

I. For some even $n$ , there is no real-valued root.

II. For any odd $n$ , there is a real-valued root.

III. For any even $n$ , there is no real-valued root.

Note: The sequence is intended to be in "decreasing" order of $n$ .