APM Advances in Pure Mathematics 2160-0368 Scientific Research Publishing 10.4236/apm.2018.86033 APM-85519 Articles Physics&Mathematics A Fundamental Relationship of Polynomials and Its Proof Serdar Beji 1 * Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, Istanbul, Turkey * E-mail:sbeji@itu.edu.tr 20 06 2018 08 06 559 563 21, May 2018 23, June 2018 26, June 2018 © Copyright 2014 by authors and Scientific Research Publishing Inc. 2014 This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

A fundamental algebraic relationship for a general polynomial of degree n is given and proven by mathematical induction. The stated relationship is based on the well-known property of polynomials that the n th - differences of the subsequent values of an n th - order polynomial are constant.

Polynomials of Degree n n<sup>th</sup>-Order Finite-Differences Recurrence Relationship for Polynomials
1. Introduction

The “Fundamental Theorem of Algebra” states that a polynomial of degree n has n roots. Its first assertion in a different form is attributed to Peter Rothe in 1606 and later Albert Girard in 1629. Euler gave a clear statement of the theorem in a letter to Gauss in 1742 and at different times Gauss gave four different proofs (see  , p. 292-306).

A nearly as important property of a polynomial is the constancy of the nth‑differences of its subsequent values. To clarify this point let us begin with some demonstrations. While it is customary to use polynomials with real coefficients, here a second-order polynomial with complex coefficients is considered first,

P 2 ( x ) = ( 1 + i ) x 2 − 3 i x + 2 (1)

where i = − 1 is the imaginary unit. Taking a real starting point x 0 = − 2 and a real step value s = 1 the following Table 1 of differences can be established for the subsequent values of the polynomial.

The first differences are computed by taking the differences of the subsequent values of the polynomial as in P 2 ( − 2 ) − P 2 ( − 1 ) = ( 6 + 10 i ) − ( 3 + 4 i ) = 3 + 6 i .

References Smith, D.E. (1959) A Source Book in Mathematics. Dover Publications, New York. Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions. Dover Publications, New York.