How to calculate the approximate zero of any polynomial function without Newton's Method? Newton's method is very interesting, however I am not always sure that I will get what I am looking for when I use it. For example, if I use it to get an approximation for $x^2 = 3$, This method will only work if the number I guess isn't at the vertex of the parabola.
And there are also other issues, for instance, I might get a "cycle".
Is there any other method which does what Newton's Method does, but which doesn't depend on me choosing an adequate point?

By the way, with regards to Newton's, Bisection and Brent's method, which is better? I mean, could you please give me the pros and cons of those methods and/or others alike?

There's the Bisection Method, but it's slow.
