As all of us know, the easiest method currently available in finding the zeroes of a polynomial is based in factorisation of polynomials.
I tried to find a faster way of calculating the zeroes of a quadratic polynomial, but ended up getting a trivial rewrite of the quadratic formula :
If $f(x) = ax^2+bx+c$, then the zeroes of the polynomial $f(x) = \frac{-b}{2a} \pm \sqrt{f(\frac{-b}{2a})\times \frac{-1}{a}} $
Looking at a linear polynomial $ax + b$, $x = \frac{-b}{a}$ is its zero.
Observing the above forms, we can see that the denominator of the zeroes get multiplied by the polynomial's degree (in both the polynomials) and an extra term comes where the multiplicative inverse (reciprocal) of the degree is its power (as in the case of the quadratic polynomial).
Now, my doubt is : can the zeroes of any polynomial be found using such forms (as given above)? Maybe some $\frac{-b}{na} (n = \text{degree of the polynomial})$ can be used to deduce the zeroes faster ? Or are all the formulae to find the zeroes of a polynomial of a specific degree (be it $2,3,4,5.... $etc.)based on such forms ?
I have only heard of the ways to calculate the zeroes of the polynomials of degree $1$ to $3$ and nothing more, since I am a tenth grader. That's why I am asking this.
