Calculadora da Regra dos Sinais de Descartes

The Descartes' Rule of Signs Calculator analyzes the polynomial P(x) = a₃x³ + a₂x² + a₁x + a₀ to determine the maximum number of positive real roots. It counts how many times the coefficients of the polynomial change sign (from positive to negative or vice versa). This number of changes indicates the upper limit of positive roots. For example, if P(x) = x³ - 2x² + x - 1, the signs are +, -, +, -, resulting in 3 changes, meaning there can be 3, 1, or no positive roots.

The rule works because each sign change corresponds to a possible positive root, but the actual number may be less by an even number. This is due to complex roots appearing in conjugate pairs. The calculator can also be used for negative roots by analyzing P(-x). For example, for P(x) above, P(-x) = -x³ - 2x² - x - 1, with 0 changes, indicating no negative roots.

Use this calculator when you need to quickly estimate how many positive real roots a cubic polynomial may have. It is useful in algebra, engineering, and physics problems where you want to know about positive roots without solving completely. For instance, when modeling population growth or radioactive decay, positive roots indicate equilibrium points.

Cautions: the rule does not give the exact number of roots, only an upper bound. Also, it does not directly identify negative or complex roots. For a complete analysis, combine with other methods like Bolzano's theorem or factorization. Remember that the rule applies to polynomials with real coefficients ordered by descending powers.