There is a simple explanation that shows why the Fundamental Theorem of Algebra can not be proved without results of Analysis? A "challenge" that graduate students often do in Algebra for students doing a first course in algebra is: "Prove the Fundamental Theorem of Algebra without using the results of analysis."
To study analysis is necessary axioms that define fields (algebraic axioms) and the supreme axiom (an axiom purely analytical).
It is well known that the supreme axiom implies the Intermediate Value Theorem. Hence it follows Rolle's Theorem, the Mean Value Theorem and the Fundamental Theorem of Calculus, and etc ...
What we can understand about this 'challenge' folkloric we can use all the axioms of fields and equipped with an algebraic construction of the rings of polynomials in one variable to prove the theorem. However we can not use the axiom of the supreme and none of its consequences.
The question that comes to mind is: would it be possible to prove the Fundamental Theorem of Ágebra without using the axiom of supreme?
Always see in books that the answer to this question is no. My question is:

There is a simple ( or intuitive ) explanation that shows why the Fundamental Theorem of Algebra can not be proved without results analysis as described above?

Thank's
 A: From the wiki page on the fundamental theorem of algebra:
"In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients."
Why is say, the proof of the fundamental theorem of algebra using Liouville's Theorem not considered simple? After all, isn't it only natural to try and use complex analysis to prove a theorem which says that a polynomial has all of it's roots in the complex plane? You can find a bunch of different proofs here. The first line in that link may be a partial answer: 
"All proofs below involve some analysis, or at least the topological concept of continuity of real or complex functions. Some also use differentiable or even analytic functions. This fact has led some to remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra."
