Descartes’ Rule of Signs Calculator (possible positive & negative real zeros)

Understanding the Descartes' rule of signs calculator — concept and principle

This calculator applies Descartes' rule of signs to a polynomial: it counts the sign changes in p(x) and p(−x) and lists the possible numbers of positive, negative and non-real zeros. Use it before solving a polynomial equation, to know what kinds of roots to look for.

How it is calculated

1. Write p(x) from the highest power down, skipping zero coefficients.
2. Count sign changes between consecutive coefficients. The number of positive real zeros equals that count or is less by an even number.
3. Form p(−x), which flips the sign of every odd-power term, and count its sign changes for the negative real zeros.
4. If the constant term is 0, factor out xᵐ first: x = 0 is a zero of multiplicity m.

• Non-real zeros come in conjugate pairs, so only combinations whose remaining count is even are listed; the counts always add up to the degree.

The principle

Each positive root r of a polynomial corresponds to a factor (x − r), and multiplying by such a factor adds at least one sign change to the coefficient list. That is why sign changes give an upper limit on positive roots. Complex roots of a polynomial with real coefficients come in pairs, so the actual count can only fall short by an even number. Substituting −x turns negative roots into positive ones, letting the same count work for them.

Worked example

Default input: p(x) = x³ − 3x² − 6x + 8.

1. Signs of p(x): + − − +, so 2 sign changes: 2 or 0 positive real zeros.
2. p(−x) = −x³ − 3x² + 6x + 8, signs − − + +: 1 sign change, so exactly 1 negative real zero.
3. The constant term is 8, so x = 0 is not a zero (multiplicity 0).
4. Possible (positive, negative, non-real): (2, 1, 0); (0, 1, 2).

How to read the result

The rule narrows the possibilities but does not pick one. Here the polynomial factors as (x − 1)(x − 4)(x + 2), so the true case is 2 positive and 1 negative zero. Combine the rule with the rational root test and a quick graph to find the roots efficiently.

Common mistakes

• Counting a zero coefficient as a sign change.
• Forgetting to change only the odd-power terms in p(−x).
• Reading "2 sign changes" as exactly 2 positive roots.
• Counting x = 0 as positive or negative.

Frequently asked questions

What does Descartes' rule of signs tell you?

The maximum number of positive and negative real zeros, and that the true numbers differ from these by even numbers.

How do I find p(−x)?

Change the sign of every term with an odd power of x.

Does it work for non-polynomial functions?

No, only for polynomials with real coefficients.

Sources: OpenStax, College Algebra 2e (2021), §5.5 Zeros of Polynomial Functions