More on the Lost Cousin of the Fundamental Theorem of Algebra

Abstract

g(x) = a0x0 + a1x1 + · · · + an xn , where α0 < α1 < · · · < αn are arbitrary real numbers and an = 0, has no more than n roots. Proof. We proceed by induction on n, noting that for n = 1 the statement is obvious. Assume that for some n the claim is true, but for n + 1, it is not. Hence, for some real numbers α0 < α1 < · · · < αn < αn+1 and an+1 = 0, there is a function g(x) = a0x0 + a1x1 + · · · + an xn + an+1xn+1 , whose number of positive roots is larger than n + 1. These roots are identical with the roots of the new function