How this instrument works
Descartes' Rule of Signs reads a polynomial's coefficients from the highest power down to the constant and counts how many times consecutive nonzero terms switch sign. For a cubic ax³ + bx² + cx + d, that means checking three neighbouring pairs: a against b, b against c, and c against d. The resulting tally is not the number of positive real roots — it is the most that number is allowed to be.
The bound follows from how real and complex roots interact inside the coefficients. Multiplying a lower-degree polynomial by (x − r) for a positive real root r always inserts at least one sign change into the sequence, while multiplying by an irreducible quadratic factor — the signature of a conjugate pair of complex roots — never removes more sign changes than it adds. Stack enough of both kinds of factor and the sign-change count can only sit at or above the true number of positive real roots, never below it.
One consequence catches people off guard: because non-real roots always arrive in conjugate pairs, the actual positive-root count shares the sign-change tally's parity exactly. Three sign changes therefore permits three positive real roots or one, never two or zero. The rule also assumes every coefficient it compares is nonzero — a zero coefficient is meant to be skipped over, not read as its own flat step with no direction to flip.
- Enter your cubic's leading number into Coefficient of x³, then the next term's number into Coefficient of x².
- Fill Coefficient of x with the linear term's number and Constant term with whatever is left when x = 0.
- Read Sign changes (max positive real roots): it counts how many of the three neighbouring pairs — x³ to x², x² to x, x to constant — flip from positive to negative or back.
- Treat that figure as a ceiling: the true count of positive real roots equals it, or two less, or four less, down toward 0 or 1.
Worked example — the cubic with roots 1, 2, and 3
Take x³ − 6x² + 11x − 6, which factors neatly as (x − 1)(x − 2)(x − 3) — its roots are exactly 1, 2, and 3, chosen so every step below can be checked by hand. Entering the coefficients gives Coefficient of x³ = 1, Coefficient of x² = −6, Coefficient of x = 11, and Constant term = −6.
Reading left to right, the signs are +, −, +, −, and every one of the three neighbouring pairs flips: a to b goes positive to negative, b to c goes negative to positive, and c to d goes positive to negative again — three sign changes in total. Sign changes (max positive real roots) accordingly reads 3, and because the cubic really does have three positive real roots, this is one of the cases where the bound is met exactly rather than merely allowed.
Questions
What does Descartes' Rule of Signs actually count?
It counts sign changes between consecutive nonzero coefficients, read from the highest power down to the constant term. For a cubic ax³+bx²+cx+d that means three pairs: a to b, b to c, and c to d. The tally is not the positive-root count itself — it is the largest that count is allowed to be, with the true value equal to it or less by an even number.
Why can the true root count be less than the sign-change count?
Because non-real roots always occur in conjugate pairs. Trading two complex roots for two real ones leaves the polynomial's degree unchanged but lowers the positive-root count by exactly two, while never changing the sign-change tally, which stays fixed as an upper limit. Three sign changes therefore permits three positive real roots or one — never two or zero.
Does a coefficient of zero count as a sign change?
Not in the rule's full statement — zero coefficients are skipped, and you compare the two nonzero terms surrounding the gap. This calculator's fields test only the three literal neighbouring pairs, a to b, b to c, and c to d, so a genuine zero coefficient should be dropped out and its neighbours compared by hand before the reading is trusted.
How is this different from actually solving for the roots?
Descartes' Rule needs no root-finding step at all — it reads a bound straight off the coefficients' signs, which makes it a fast check before reaching for the cubic formula or a numeric solver. It never reports where the roots sit, only how many positive ones are possible, which is often enough to catch a factoring mistake or confirm a graph's crossings.
Does the rule say anything about negative real roots?
Yes, with one extra step: substitute −x for x throughout the polynomial, which flips the sign of every odd-power term, then count sign changes in the new sequence. That flips a and c — the x³ and x coefficients — while b and d stay as entered; re-enter the four adjusted numbers to read the negative-root bound from this same sheet.
Who is credited with this result, and how old is it?
René Descartes stated the rule in 1637 in La Géométrie, an appendix to his Discourse on the Method, though without a complete proof — that arrived later as other mathematicians, including Gauss, tightened the argument. It remains a purely algebraic bound: no calculus, no iteration, just the coefficients' signs.