Descartes' Rule of Signs Calculator

Analyze Positive and Negative Real Roots of Polynomials

Descartes’ Rule of Signs Formula

Formula

\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \\[1em] V(x) = \text{number of sign changes in the coefficients of } P(x) \\[1em] N_{\text{pos}} \le V(x), \quad N_{\text{pos}} \equiv V(x) \pmod{2} \\[1em] N_{\text{neg}} \le V(-x), \quad N_{\text{neg}} \equiv V(-x) \pmod{2} \]

Where:

\(P(x)\) — the polynomial, with coefficients written in descending order of powers
\(V(x)\) — the number of sign changes between consecutive nonzero coefficients of \(P(x)\)
\(V(-x)\) — the number of sign changes in \(P(-x)\), i.e. the polynomial with the signs of all odd-power coefficients flipped
\(N_{\text{pos}}\), \(N_{\text{neg}}\) — the number of positive and negative real roots, counted with multiplicity

The rule states that the number of positive real roots equals the number of sign changes in \(P(x)\) or is less than it by an even number. The same applies to negative roots after substituting \(-x\) for \(x\).

Descartes’ Rule of Signs – Calculation Example

Example: take the polynomial \(f(x) = x^3 - 6x^2 + 11x - 6\), entered as coefficients 1 -6 11 -6.

Step 1 — positive roots. The signs of the coefficients of \(f(x)\) are:

\[ +,\ -,\ +,\ - \quad \Rightarrow \quad 3 \text{ sign changes} \]

So there are exactly 3 or exactly 1 positive real roots.

Step 2 — negative roots. Substitute \(-x\):

\[ f(-x) = -x^3 - 6x^2 - 11x - 6 \]

The signs are \(-, -, -, -\), giving 0 sign changes, so there are no negative real roots.

Result: 3 or 1 positive roots, 0 negative roots, and at least \(3 - 3 = 0\) non-real roots. In fact \(f(x) = (x-1)(x-2)(x-3)\), so all three roots (1, 2, 3) are positive.

How to use the calculator

In the Polynomial Coefficients field, type the coefficients of your polynomial in descending order of powers, separated by spaces. For example, 1 -6 11 -6 stands for \(x^3 - 6x^2 + 11x - 6\).
Include zero coefficients where a power is missing: 2 0 -5 1 means \(2x^3 - 5x + 1\). Leading zeros and trailing zeros are handled automatically.
Press Calculate.
Read the results: the calculator shows your polynomial in standard form, the multiplicity of zero as a root (if the constant term is 0), the possible numbers of positive real roots, the possible numbers of negative real roots, and the minimum number of non-real (complex) roots.

What Descartes’ Rule of Signs tells you

Descartes’ Rule of Signs, published by René Descartes in 1637, is a quick test that bounds how many positive and negative real roots a polynomial can have — without actually solving the equation. You write the polynomial in standard form and count how many times the consecutive nonzero coefficients switch sign. That count, reduced by an even number, is the exact number of positive roots. Repeating the count on \(f(-x)\) gives the possibilities for negative roots.

The “differ by an even number” part matters. If there are 3 sign changes, the polynomial has 3 or 1 positive roots — never 2 or 0 is not guaranteed; only 3 or 1 are possible. If there is exactly 1 sign change, the polynomial has exactly 1 positive root, which is one of the most useful special cases of the rule.

Combining the two counts with the degree of the polynomial also tells you the minimum number of non-real roots. Since a polynomial of degree \(n\) has exactly \(n\) roots in total (counting multiplicity, by the Fundamental Theorem of Algebra), and non-real roots always come in conjugate pairs, subtracting the maximum possible counts of real roots from the degree gives the minimum number of complex roots. To explore those, see the Complex Root Calculator and the Complex Conjugate Calculator.

Interpreting the results

Each result line lists every possibility consistent with the rule. “3 or 1” for positive roots means both outcomes are compatible with the sign pattern; additional information (a graph, rational root tests, or numerical methods) is needed to pin down the actual count. A result of exactly 0 is definitive: if \(f(-x)\) has no sign changes, the polynomial has no negative real roots, period.

If the constant term is zero, \(x = 0\) is a root, and the calculator reports its multiplicity by counting trailing zero coefficients. In that case, apply the rule to the remaining part of the polynomial after factoring out \(x^k\).

Assumptions, limitations and tips

Coefficients must be real. The rule does not apply to polynomials with complex coefficients.
Zero coefficients are skipped when counting sign changes. For \(x^3 + 1\), the signs are \(+, +\) — one sign change, hence exactly 1 negative root after checking \(f(-x)\) as well.
The rule counts multiplicity. A double root at \(x = 2\) counts as two positive roots.
It gives possibilities, not exact values. The rule never tells you what the roots are, only how many of each sign can exist. Use it to narrow the search before applying other tools.
Zero roots give a quick check: if a polynomial of odd degree has no sign changes in \(f(x)\), it must have at least one negative root, because odd-degree polynomials always have at least one real root.
The rule works especially well alongside equation solvers such as the Cubic Equation Calculator and the Quartic Equation Calculator, and it can confirm the root structure found with the Quadratic Equation Calculator.

Frequently Asked Questions

How do I apply Descartes' Rule of Signs by hand?

Write the polynomial in standard form with terms ordered from highest to lowest power, list the signs of the nonzero coefficients, and count how many times the sign switches between consecutive coefficients. That number, or that number minus an even number, is the possible count of positive real roots. Then substitute −x for x, simplify, count sign changes again, and you get the possibilities for negative real roots. Skip any zero coefficients when counting.

Why does the number of roots differ from the sign changes by an even number?

Non-real roots come in conjugate pairs, and roots crossing through zero change the sign pattern in pairs. So each pair of positive roots that "disappears" (becomes complex or negative) removes exactly 2 from the count. That is why 4 sign changes means 4, 2, or 0 positive roots — never 3 or 1. The parity of the sign-change count always matches the parity of the actual number of positive roots.

Does one sign change always mean exactly one positive root?

Yes — this is the most powerful special case of the rule. If the coefficients of a polynomial have exactly one sign change, it must have exactly one positive real root (counted with multiplicity). Unlike the general case, which only lists possibilities, a single sign change gives a definitive answer. It is often used to prove that an equation has at least one positive solution, for example in economics or optimization problems.

How do zero coefficients affect the count?

Zero coefficients are simply skipped — only nonzero coefficients are compared. For example, in \(x^3 + 1\) the signs are \(+, +\) for the terms \(x^3\) and \(1\), giving 0 sign changes, so there are no positive roots. If the constant term is zero, factor out the highest power of \(x\) first (the calculator reports its multiplicity), then apply the rule to what remains.

Can the rule tell me the exact number of real roots?

Not always. It narrows the possibilities to a list that differs by even numbers, such as "3 or 1 positive roots." However, some results are definitive: 0 sign changes means exactly 0 roots of that sign, and 1 sign change means exactly 1. To pin down ambiguous cases, combine the rule with a graph, the rational root test, or numerical methods.

How is the minimum number of complex roots calculated?

A polynomial of degree \(n\) has exactly \(n\) roots in total, counting multiplicity. Take the maximum possible positive roots (sign changes in \(f(x)\)) plus the maximum possible negative roots (sign changes in \(f(-x)\)), add the multiplicity of zero if any, and subtract from the degree. The remainder is the minimum number of non-real roots, always even, since complex roots come in conjugate pairs.

What are common mistakes when using the rule?

The most frequent errors are: forgetting to write the polynomial in descending order of powers before counting, including zero coefficients in the count (they should be skipped), forgetting that roots are counted with multiplicity (a double root counts twice), and miscomputing \(f(-x)\) — only the odd-power coefficients flip sign, so \(x^2\) and the constant term stay the same. Also remember the rule requires real coefficients.

Descartes’ Rule of Signs Calculator