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Descartes's Rule of Signs

The next question to be tackled was how to predict the number of positive, negative, and complex roots. Cardan noted that the complex roots of an equation occur in pairs. This is the conjugate pair theorem used today. Newton went on to prove this in his Arithmetica Universalis. Descartes stated his rule on signs in La Géométrie, but did not give a proof. He said that the maximum number of positive roots of f(x) = 0, where f is a polynomial, is the number of changes in sign of the coefficients and that the maximum number of negative roots is the number of sign changes in f(-x) = 0. This became known as Descartes's rule.

There is disagreement whether the rule of signs was known before Descartes's La Géométrie. Some say Thomas Harriot gave it in his Artis analyticae praxis, published in 1631. Cantor denies this since Harriot didn't admit negative roots. Cardan also stated a relation between one or two variations in sign and the occurrence of positive roots. Newton produced a procedure for finding the number of imaginary roots. Leibniz suggested an outline of a proof, but did not give a detailed one. Several mathematicians went on to prove and refine Descartes' rule from 1745-1828.

In 1828 Gauss added a statement to the rule. He said that if the number of positive roots falls short of the number of variations of sign, it does so by an even integer. This is also true for negative roots.

In summary, Descartes' rule of signs is a tool to help determine what type of roots, positive, negative, or imaginary, a polynomial equation has.