Which branches of maths study the ways solving polynomial equations? I mean , for example , let 
$$0 = 1 + 2x + 3x^{2} +x^{7} + 19 x^{9 }$$
and we want to solve this equation, what branch of maths doing this? 
I know that there is no algebraic solution of the problem but there is methods like Newton's for instance . 
What is the branch which look for these numerical methods for solving equations? 
What are the pre-requisite subjects required for this branch? 
 A: I think your question is very broad as it depends on context.
For example, the following branches of mathematics study solving polynomial equations:
Abstract Algebra
Algebraic Geometry
Numerical Analysis
Number Theory
Would you include others too? Certainly depending on context.
Update
If you are interested in a book in Numerical Analysis, the book Numerical Analysis by Richard L. Burden and J. Douglas Faires is excellent (wow - look at the price). They even have a web site where there is code in SW like Maple and Mathematica.
Of course, there are also some Dover (great prices), like Introduction to Numerical Analysis: Second Edition by F. B. Hildebrand.
You can also find many things online for free like Equation Solving.
You can also find Open Course-ware at place like MIT where you can find notes and video lectures on the topic.
Lastly, you might look into a Computer Algebra System - which include open source freeware and commercial variants. The offer many routines for this which are optimized or you can program your own (like the routines by Burden and Faires) routines to experiment with methods, their pitfalls, error analyses and the like.
Regards
A: In general when there's no algebraic solution, as your tags imply, numerical methods are used.
(If this it's a Diophantine equation, it has no solutions as the only candidates are +-19)
A: Numerical analysis is probably what you're looking for. That would include methods for finding zeros of polynomials and other complex functions.
Numerical analysis is a pretty broad subject, so it relies on lots of other areas of mathematics. In the particular area you're asking about, results from abstract algebra and linear algebra are helpful, among others.
I'd suggest you go flip through a few numerical analysis books, and see what kinds of techniques they use. 
