Cyclic code with generator polynomial

Question

Response to the following problem:

Let s(X) denote the syndrome corresponding to error sequence e(X) in an (n, k) cyclic code with generator polynomial g(X). Show that the syndrome corresponding to e⁽¹⁾(X), the right cyclic shift of e(X), is s⁽¹⁾(X), defined by

s⁽¹⁾(X) = Xs(X)  mod g(X)