In this paper we are going to study the polynomials whose arguments and coefficients are vectors in the Euclidean vector space together with the new operations defined in our previous paper viXra:2510.0152. In order to prove the Fundamental Theorem of Algebra with topological tools, we are going to define the limits and derivatives with respect to vectors. We are going to represent some values of the polynomials thanks to paths in the plane. We will see that for the partial sums of the Taylor development linked to the exponential function, we get spiral paths leading to the unit circle. We are going to find the zeros/roots and we are going to present a new formulation of the Fundamental Theorem of Algebra, in the Euclidean vector space, with its meaning linked to paths in the plane. We are going to adapt a proof made by Laurent Schwartz with complex numbers. We are going to present also an adaptation of the algorithm of Kneser in order to find the roots. We will show that the Fundamental Theorem of Algebra is definitely geometrical. We will give also a link to a code source for GNU Octave for experiments with operations and polynomials in this framework.