Refresher - Descartes' Rule of Signs

Descartes' rule of signs

Positive roots

The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less than it by a multiple of 2. Multiple roots of the same value are counted separately.

Negative roots

As a corollary of the rule, the number of negative roots is the number of negative integers after negating the coefficients of odd-power terms (otherwise seen as substituting the negation of the variable for the variable itself), or fewer than it by a multiple of 2.

Example

For example, the polynomial ${\displaystyle x^3+x^2-x-2}$

has one sign change between the second and third terms. Therefore it has exactly one positive root. Note that the leading sign needs to be considered although it doesn't affect the answer in this case. In fact, this polynomial factors as ${\displaystyle x^3+x^2-x-2}$

so the roots are −1 (twice) and 1.

Now consider the polynomial ${\displaystyle -x^3+x^2+x-1}$

This polynomial has two sign changes, meaning the original polynomial has two or zero negative roots and this second polynomial has two or zero positive roots. The factorization of the second polynomial is

${\displaystyle {(x-1)}^2(x+1)}$,

So here, the roots are 1 (twice) and −1, the negation of the roots of the original polynomial. Since any nth degree polynomial equation has exactly n roots, the minimum number of complex roots is equal to

${\displaystyle (n-(p+q))}$,

Where p denotes the maximum number of positive roots, and q denotes the maximum number of negative roots (both of which can be found out using Descarte's rule of sign), and n denotes the degree of the equation.