At the moment we just learnt the factor theorem of polynomials and how if $x-a$ is a factor of $P(x)$, then $P(a) = 0$. We're then taught to find the roots of a polynomial its best to check the factors of it's constant and see what happens when it's factors are plugged into the polynomial. If you get 0, good job and divide the polynomial, and repeat the process. I feel like this is a bit like being taught factorisation for quadratics, its a good and all, until you learn the quadratic formula and it can solve all quadratics!
How would I find the roots of any non constant, single variable polynomial?
I'm guessing there's unlikely to be an equation for such a task, and I'm uncertain of how it would be done. If it can be done at all.