Descartes' Rule of Signs Calculator

Descartes' rule of signs relates the number of sign changes in the coefficients of a polynomial in standard form to the number of its positive and negative real roots. First, arrange your polynomial in descending order of powers: P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0. Count the number of sign changes between consecutive non-zero coefficients. This count, denoted as v, represents the maximum possible number of positive real roots. Furthermore, the actual number of positive roots is either equal to v or less than v by an even integer (due to complex conjugate pairs). To find negative real roots, evaluate P(-x) and count the sign changes in that modified polynomial using the exact same logic.

Consider the polynomial P(x) = 2x^4 - 5x^3 + 2x^2 - x + 3. Let us determine the potential positive and negative real roots step by step. First, examine the signs of the coefficients in order: +2, -5, +2, -1, +3. We count the changes from positive to negative and vice versa: from +2 to -5 is one change, from -5 to +2 is a second change, from +2 to -1 is a third change, and from -1 to +3 is a fourth change. There are 4 sign changes, meaning P(x) has either 4, 2, or 0 positive real roots. Next, substitute -x into the polynomial to find P(-x) = 2(-x)^4 - 5(-x)^3 + 2(-x)^2 - (-x) + 3, which simplifies to 2x^4 + 5x^3 + 2x^2 + x + 3. Looking at the coefficients here (+2, +5, +2, +1, +3), there are 0 sign changes. Therefore, the polynomial has exactly 0 negative real roots.

Always ensure your polynomial is fully expanded and written in standard descending order before counting sign changes. Remember to skip any coefficients that equal zero, as they do not contribute to a sign change. Keep in mind that this rule provides an upper bound and accounts for possible complex roots, meaning you may need synthetic division or graphical methods to find the exact root values.