Is it hard to find the root compare to showing its existence? Fundamental theorem of Algebra : It says that a polynomial over complex numbers of degree $d$ has at least one root complex root.
My Question : Is it hard to find the root compare to showing its existence ?
 A: It depends on what you mean by find.  I expect that you know the formula for roots of a quadratic (degree 2) polynomial.  You probably learned it before you learned the complex numbers; it applies in the complex case as well but with the difference that you can always perform the square root.  
There is also a formula for the cubic (degree 3) case but it is much more complicated and not so well known.  Cubic function at Wikipedia.  An interesting feature of this is that complex numbers may occur in the calculations even if the result is real.  
There is an even more complicated formula for the quartic (degree 4) case: Quartic function at Wikipedia.  
You might expect the trend to continue but it does not. There is no general solution in radicals (normal arithmetic and roots).  Not just not known but provably not possible.  Quintic formula at Wikipedia.  
So, in a sense, yes it is hard to find the roots of an arbitrary polynomial (real or complex).  
On the other hand, numerical methods will easily calculate the roots of any polynomial to any desired accuracy.  So, it is easy.  
