Descartes' Rule of Signs Calculator

Enter a polynomial expression (e.g. x^3 - 2x^2 + x - 5) and optionally specify the variable, then apply Descartes' Rule of Signs to determine the maximum number of positive real roots, negative real roots, and imaginary roots. The tool parses your polynomial, counts sign changes in the coefficient sequence, and returns a complete possibilities table showing every valid combination of root types.

Use ^ for exponents (e.g. x^3 - 2x^2 + x - 5). Supported variable: x.

Results

Max Positive Real Roots

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Max Negative Real Roots

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Sign Changes (f(x))

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Sign Changes (f(-x))

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Possible Root Combinations

Results Table

Frequently Asked Questions

What is Descartes' Rule of Signs?

Descartes' Rule of Signs is a mathematical theorem that gives an upper bound on the number of positive and negative real roots of a polynomial with real coefficients. The number of positive real roots equals the number of sign changes between consecutive nonzero coefficients, or is less than that count by an even number. A similar rule applies to negative roots by substituting -x into the polynomial.

Why do we use Descartes' Rule of Signs?

The rule is used to narrow down the possible number of real roots before applying more complex root-finding methods. It helps mathematicians and students quickly assess the structure of a polynomial's roots without fully solving it, saving significant time during analysis.

How do I find the number of negative real roots?

To find negative real roots, substitute -x for every x in your polynomial to get f(-x), then simplify and count the sign changes between consecutive nonzero coefficients of the resulting polynomial. The number of negative real roots equals that sign-change count or is less by an even number.

What is a real root?

A real root (or real zero) of a polynomial is a value of x that makes the polynomial equal to zero, and that value is a real number (not imaginary). For example, x = 2 is a real root of x^2 - 4 because substituting gives 4 - 4 = 0.

Can a polynomial have only imaginary roots?

Yes. If there are no sign changes in f(x) and no sign changes in f(-x), the polynomial has zero positive and zero negative real roots. In that case, all roots are imaginary (complex). For example, x^2 + 1 has two imaginary roots: i and -i.

Why make a Descartes' Rule of Signs possibilities table?

Because the rule gives a maximum count and allows reductions by even numbers, multiple combinations of positive, negative, and imaginary roots are possible. A table systematically lists every valid combination, ensuring no case is overlooked when analysing the polynomial.

What are the limitations of Descartes' Rule of Signs?

The rule provides an upper bound and possible counts, but does not give the exact number of roots or their actual values. It also does not account for complex (non-real) roots directly — those are inferred from the degree minus the real roots. To find exact roots, you still need methods like factoring, synthetic division, or numerical solvers.

How does the calculator handle repeated roots or zero coefficients?

Terms with a coefficient of zero are skipped when counting sign changes, as per the rule — only nonzero coefficients matter. Repeated (multiplicity > 1) roots are counted according to their multiplicity in Descartes' framework, so a double root counts as two roots of the same sign.

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