Fundamental Theorem of Algebra

Introduction:

         The Fundamental Theorem of Algebra is a for equation solving. It means that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution. Polynomial equations are in the form

                                               P(x) = a_nxⁿ + a_n-1 x ^n-1 + ... + a₁x + a₀ = 0,
where an is assumed non-zero in which case n is called the degree of the polynomial P and of the equation above.

Fundamental Theorem of Algebra:

       A number a is a solution to the equation P(x) = 0 if substituting a for x makes it identity: P(a) = 0. The coefficients are assumed to belong to a exact set of numbers where we also look for a solution. The polynomial form is very general but frequently studying P(x) = Q(x) is more suitable.
      To see how it works let's start with the counting numbers (N=numbers 1,2,3,...), and the simplest equation             x + a = b. For example, x + 5 = 12 has a solution x = 12 - 5 = 7. Introduction of negative numbers eases the problem:
Any equation x + a = b where a, b belongs to N has a solution x belongs to Z, where Z is the set of integers numbers like plus, minus whole numbers and zero.
Once we accepted the negatives, we have a stronger result:
Any equation x + a = b where a, b belongs to Z has a solution x belongs to Z.
Even if the coefficients are permitted to be negative, the equation still has a solution in Z. Now let's consider other equations over Z: 11x + 11= 0, x= -11 /11 = -1. 
   Any equation ax + b = 0 where a,b belongs to z has a solution x belongs to Q.
Moreover, we again get a stronger result,
Any equation ax + b = 0 where a,b belongs to Q has a solution x belongs to Q.
       Hope you liked the above explanation. Please leave your comments, if you have any doubts.