I have developed an interest in generalisations of the fibonacci sequence, from tribonacci sequence up to what I'll coin the 'infinibonacci' sequence.

I'm aware that these nth-bonacci sequences require roots of matched nth-degree polynomials in the form $x^n - (x^{n-1} + x^{n-2} + \dots +x^0)=0$ and that 5th-degree polynomials are not guaranteed to be solvable, but I'm wondering if there is a systematic way of showing whether these particular polynomials are solvable by root extraction?

I am prepared to learn anything required to understand the result but am currently starting from smart 2nd year undergraduate level.
Thanks.