We define the multiplicity of a root \(r\) to be the number of factors the polynomial has of the form \(x - r\). The number of positive real roots either is equal to the number of variations in the si...We define the multiplicity of a root \(r\) to be the number of factors the polynomial has of the form \(x - r\). The number of positive real roots either is equal to the number of variations in the sign of \(P(x)\) or is less then that by an even integer. Alternatively, to compute the number of negative roots, you change the sign of all the odd terms of \(P(x)\) and count the number of signs.
