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.
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 [
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
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 .