Abstract
The modern formulation of Descartes' Rule of Signs provides a list of possible numbers of both positive and negative roots for a given sign sequence. We have shown (assuming none of the signs are zero) that polynomials exist with any of the possible numbers of positive roots. By replacing x by -x, we can provide a polynomial with the given sign sequence that contains any given number of negative roots allowable by Descartes. We have not addressed trying to accommodate both the positive and negative roots simultaneously. The following question is unanswered here and seems to be open: Given a sign sequence (which may include some zeros), do there exist polynomials containing positive and negative roots numbering each of the possible combinations allowed by Descartes' Rule of Signs?
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