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Descartes' Rule of Signs gives a quick bound on how many positive and negative real roots a polynomial can have, just by counting how many times consecutive coefficients switch sign — no root-finding required.
How it works
The number of sign changes between consecutive non-zero coefficients of p(x) gives the number of positive real roots, or that number minus a multiple of 2. Substituting -x for x (flipping the sign of every odd-degree term) and repeating the count gives the same information for negative real roots.
- Enter x⁵ coefficient (optional).
- Enter x⁴ coefficient (optional).
- Enter x³ coefficient (optional).
- Enter x² coefficient.
- Enter x coefficient.
- Enter constant.
- Click Calculate to see your results.
Examples
A cubic example
p(x) = x³-1 has coefficients 1, 0, 0, -1 — one sign change, so exactly 1 positive real root. p(-x) = -x³-1 has no sign changes, so 0 negative real roots.
A polynomial with more possibilities
p(x) = x³-2x²-x+2 has 2 sign changes, so 2 or 0 positive real roots — the rule gives a range, not always an exact count.
Who should use it
- Precalculus and algebra coursework on polynomial root analysis.
- A quick sanity check before attempting to fully factor or solve a polynomial.
Industry applications
- Mathematics education
Advantages
- Works for any polynomial up to degree 5 with no root-finding computation needed.
- Gives bounds on both positive and negative real roots in one pass.
Limitations
- Only bounds real root counts — it says nothing about complex roots directly, and often gives a range rather than an exact count.
Common mistakes to avoid
- Counting a zero coefficient as a "sign" when checking for changes — zero coefficients are simply skipped, not treated as a sign change.
Best practices
- Combine this rule with the Rational Zeros Calculator: Descartes' rule narrows down how many roots to look for, while the Rational Root Theorem narrows down what they might be.
Tips
- The total of the maximum positive count, maximum negative count, and number of complex roots always equals the polynomial's degree.