What this calculator does
Descartes' rule of signs puts an upper bound on how many positive real roots a polynomial can have, without solving anything. Read the coefficients from the highest power down and count the places where the sign flips. That count is the maximum. The true number is either the count itself or less by an even amount, so three sign changes allows three positive roots or one, never two.
Substituting −x for x and counting again gives the same kind of bound for negative roots. The rule will not tell you what the roots are, and it will not say which of the allowed counts is the right one, but it narrows the field before any solving starts and it costs nothing beyond reading a list of signs.
The formula
The coefficients are read in order from the highest power down, with zero coefficients skipped because they carry no sign. Every place where the sign flips counts as one change, and the total is the maximum number of positive real roots. Replacing x with −x flips the sign of every odd-power term, and counting the changes in that version gives the maximum number of negative real roots. Each count can then be reduced by 2 repeatedly, which is why the answers are given as a list of possibilities rather than a single number.
| Term | Meaning |
|---|---|
| Sign change | A place in the coefficient list where a positive is followed by a negative or the reverse, ignoring any zeros in between. |
| p(−x) | The polynomial with x replaced by −x, which flips the sign of every term whose power is odd. |
| Reduction by two | The allowance that the real count may fall short of the sign-change count by any even number, because complex roots arrive in conjugate pairs. |
| Degree | The highest power in the polynomial, and the total number of roots once complex ones are counted. |
The inputs explained
| Field | What to enter |
|---|---|
| Coefficients, highest degree first (e.g. 1,-6,11,-6 for x³−6x²+11x−6) | The coefficients in order from the highest power down, separated by commas. Include zeros for any missing powers, so x³ − 1 is entered as 1, 0, 0, -1. |
When to use it
Narrowing a root search before solving
Knowing that a cubic can have no negative roots at all tells a numerical solver, or a person sketching the curve, where there is no point looking. The rule costs one pass over the coefficients and can rule out half the number line.
Sanity-checking an answer
If a solver returns two positive roots but the coefficients only allow three or one, something has gone wrong in the working. The rule is a cheap independent check on a result reached some other way.
Working through a polynomial theory exercise
The rule appears in most algebra courses alongside the rational root theorem and the fundamental theorem of algebra, and the counts here can be checked against the intermediate steps of a hand-worked answer.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What does the rule of signs give for different polynomials?
Five polynomials chosen to show the different shapes the answer can take.
| Coefficients | Possible positive real roots | Possible negative real roots | Sign changes in p(x) | Sign changes in p(-x) |
|---|---|---|---|---|
| 1, -6, 11, -6 | 3 or 1 | 0 | 3 | 0 |
| 1, 6, 11, 6 | 0 | 3 or 1 | 0 | 3 |
| 1, 0, 0, -1 | 1 | 0 | 1 | 0 |
| 1, -1, 1, -1, 1 | 4 or 2 or 0 | 0 | 4 | 0 |
| 1, 2, -5, -6 | 1 | 2 or 0 | 1 | 2 |
Questions
Does the rule tell me what the roots actually are?
No. It only bounds how many real roots of each sign there can be. Finding the roots themselves needs factoring, the quadratic or cubic formula, or a numerical method. The rule is a filter applied before that work, not a substitute for it.
Why can the count only drop by an even number?
Because any roots that are not real come in complex conjugate pairs, so they are removed from the real count two at a time. Losing a single root on its own is impossible, which is what keeps the possibilities spaced two apart.
What happens to zero coefficients?
They are skipped. A zero has no sign, so it cannot begin or end a sign change; the comparison is made between the non-zero coefficients either side of it. Entering the zeros is still necessary, because their positions set the powers of the remaining terms.
The result gives several possibilities. How do I pick one?
The rule on its own cannot. Narrowing further means actually examining the polynomial, by sketching it, testing values at a few points to find sign changes in the output, or solving it outright.
To find the roots of a cubic outright, see the cubic equation solver. To test a candidate root and divide it out, see the synthetic division calculator.