Methods of approximating polynomial roots

Question

I am in high school and currently studying polynomials. This includes factorisation, finding zeroes and graphing. However I have realised that all the polynomials given to us usually have discrete zeroes ( such as $1,-1,2,-2$ and so on ). This makes them quite easy to find using the factor remainder theorem. However if a polynomial has roots which are not discrete ($\sqrt2,2.243$) it can be quite difficult, maybe even impossible to find their roots using the theorem. I have heard of two other methods of approximating the roots: The Newton Method and The Method Of Bisection. If I am correct, these two methods do not return exact values and take some time to implement on large polynomials. Does there exist a more efficient method which can return exact values and in turn allow for the factorisation, by hand, of a difficult polynomial? Any input is much appreciated. :)