Descartes' Rule of Signs Calculator

The Descartes' Rule of Signs Calculator finds how many positive and negative real roots a polynomial can possibly have, without solving it. Enter the coefficients from the highest degree down to the constant term, and it counts the sign changes, lists every possible root count, and builds a full table of (positive, negative, complex) root combinations consistent with the polynomial's degree.

It's built for algebra and precalculus students sanity-checking a root count before factoring or applying the quadratic/cubic formula, and for anyone who wants a quick bound on how many real roots to expect.

How to use it

1. Enter the coefficients, comma or space separated, from the highest degree term down to the constant. For example, 1,-3,2,5,-1 represents x⁴ - 3x³ + 2x² + 5x - 1.
2. Read the sign change counts for f(x) and f(-x).
3. Check the possible root counts and the Root Combinations table for every valid (positive, negative, complex) split.
4. Use the Step-by-Step Derivation to see exactly how each count was reached.

Results update automatically as you type.

What is Descartes' Rule of Signs?

For a polynomial with real coefficients, written from the highest degree to the constant term:

• The number of positive real roots (counted with multiplicity) is either equal to the number of sign changes between consecutive nonzero coefficients of f(x), or less than that by an even number.
• The number of negative real roots follows the same rule applied to f(-x) — the polynomial with every odd-degree term's coefficient negated.

Zero coefficients are skipped entirely when counting sign changes — they don't count as a "sign" one way or the other, so a term like x⁴ + 0x³ - 1 is read as +, - (one sign change), not +, 0, -.

Why "or less by an even number"?

The rule gives an upper bound, not an exact count, because non-real roots always come in complex-conjugate pairs. If the sign changes suggest up to 3 positive roots, the real count could be 3 or 1 — the "missing" 2 would be a complex-conjugate pair instead. That's why the calculator lists every possibility (3, 1) rather than a single number, and why the Root Combinations table always fills the remainder of the degree with an even number of complex roots.

Reading the Root Combinations table

Each row is one mathematically consistent way to account for every root of the polynomial: a count of positive real roots, a count of negative real roots, and a count of complex (non-real) roots — the three always add up to the polynomial's degree. A degree-4 polynomial with 3 possible positive sign changes and 1 negative sign change, for example, could have 3 positive + 1 negative + 0 complex roots, or 1 positive + 1 negative + 2 complex roots — both rows appear in the table, and the actual roots (found by solving the polynomial) will match exactly one of them.

To actually solve for the roots rather than just bound how many there are, see the Cubic Equation Solver for degree 3 or the Quadratic Formula Solver for degree 2.

Privacy

This calculator runs entirely in your browser. Your coefficients are never uploaded, logged, or stored — the computation happens locally on your device.