Descartes's rule of signs

In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for determining an upper bound on the or negative real roots of a polynomial. It is not a complete criterion, because it does not provide the exact number of positive or negative roots.

The rule is applied by counting the number of sign changes in the sequence formed by the polynomial's coefficients. If a coefficient is zero, that term is simply omitted from the sequence.

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 an even number. Multiple roots of the same value are counted separately.

As a corollary of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than it by an even number. This procedure is equivalent to substituting the negation of the variable for the variable itself. For example, to find the number of negative roots of ${\displaystyle f(x)=ax^{3}+bx^{2}+cx+d}$, we equivalently ask how many positive roots there are for ${\displaystyle -x}$ in ${\displaystyle f(-x)=a(-x)^{3}+b(-x)^{2}+c(-x)+d=-ax^{3}+bx^{2}-cx+d\equiv g(x).}$ Using Descartes' rule of signs on ${\displaystyle g(x)}$ gives the number of positive roots ${\displaystyle x_{i}}$ of g, and since ${\displaystyle g(x)=f(-x)}$ it gives the number of positive roots ${\displaystyle (-x_{i})}$ of f, which is the same as the number of negative roots ${\displaystyle x_{i}}$ of f.

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