How this calculator works
Count sign changes between consecutive nonzero coefficients in P(x). The number of positive real zeros, counting multiplicity, is that count or a smaller non-negative number differing by an even integer. Apply the same rule to P(-x) for negative zeros. Factors of x are removed first to count zero roots separately.
Formula
Positive zeros: V(P), V(P)-2, V(P)-4, ... >= 0. Negative zeros: V(P(-x)), V(P(-x))-2, ... >= 0.
Worked example
The signs of x^3 - 6x^2 + 11x - 6 are +, -, +, -, giving 3 changes. Positive-root counts may be 3 or 1. All coefficients of P(-x) are negative, so there are no negative roots.
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Frequently asked questions
Do zero coefficients count as sign changes?
No. Skip them and compare the nearest nonzero coefficients.
Does the rule locate a root?
No. It limits how many roots may have each sign. Use rational-zero testing or a root solver to find values.
Are repeated roots counted?
Yes. The rule counts roots with multiplicity, so a positive double root contributes two.