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?

Brent's method is a commonly used one:
http://en.wikipedia.org/wiki/Brent%27s_method
From Wikipedia:
In numerical analysis, Brent's method is a complicated but popular root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less reliable methods. 
