Here you are asked to prove the Fundamental Theorem of Algebra a different way by using Rouché's Theorem. Where n E N, consider the polynomial n-1 Pn (z)z" k-0 Using the circular contour C-[z...
Please prove the theorems, thank you 6.1 Theorem. Let anx+an-1- +ag he a polynomial of degree n0 with integer coefficients and assume an0. Then an integer r is a Poot of (x) if and only if there exists a polynomlal g(x) of degree n - with integer coeficients such that f(x) (x)g(x). This next theorem is very similar to the one above, but in this case (xr)g(x) is not quite equal to f(x), but is the same except for the...
Prove that any polynomial anzn + an−1zn−1 + · · · + a1z + a0 with coefficients ai ∈ Cand degree n > 0 has at least one zero in C. You may use the bijection [S1, S1] ∼= Z that associates the homotopy class of a map with the winding number of the map.