The fundamental Theorem of Algebra

This video illustrates a proof of the Fundamental Theorem of Algebra: “Every polynomial degree at least 1 with complex coefficients has a least one complex zero“. Proof: Let P(z)= z^k a_k-1 z^k-1 ... a_1 z a_0 be any polynomial (in the video P has degree 3). Substituting z=r exp(2πt) we obtain for fixed r a continuous map from the unit circle to C. For the special polynomial z^k the image of this map winds k times around the origin. Using the deformation z^k λ(P(z)-z^k) we can continuously deform the image of zk into the image of P(z). Now consider the map of the unit disk D to C given by P (green areas in the video). If we can prove that 0 lies inside the image of D, we have proven that P has a zero.