A Fundamental Relationship of Polynomials and Its Proof

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^th-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 [1] , p. 292-306).

A nearly as important property of a polynomial is the constancy of the n^th‑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

1. Smith, D.E. (1959) A Source Book in Mathematics. Dover Publications, New York.

2. Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions. Dover Publications, New York.