A New Approach to Polynomial Identities

Using the formal derivative idea, we give a generalization for the Cauchys Theorem relating to the factors of (x + y)ⁿ–xⁿ– yⁿ. We determine the polynomials A(n, a, b) and B(n, a, b) such that the polynomial can be expanded, for any natural number n, in terms of the polynomials x+y and ax²+bxy + ay². We show that the coefficients of this expansion are intimately related to the Fibonacci, Lucas, Mersenne and Fermat sequences. As an application, we give an expansion for as a polynomial in x+y and (xz –yt)(xt–yz). We use this expansion to find closely related identities to the sums of like powers. Also, we give two interesting expansions for the polynomials and xⁿ+yⁿ that we call Fibonacci expansions and Lucas expansions respectively. We prove that the first coefficient of these two expansions is a Fibonacci sequence and a Lucas sequence respectively and the other coefficients are related sequences. Finally we give a generalization for all the previous results.