Descartes' Rule of Signs Calculator

Enter a polynomial p(x) to apply Descartes' Rule of Signs: count sign changes in p(x) and p(-x) to bound its positive, negative, and non-real complex roots.

Quick Facts

Input format
Descending powers of x
e.g. x^3 - 2x^2 + 5; missing terms are treated as zero coefficients.
Positive roots
V, V-2, V-4, ... down to 0 or 1
V = sign changes between consecutive nonzero terms of p(x).
Negative roots
W, W-2, W-4, ... down to 0 or 1
W = sign changes between consecutive nonzero terms of p(-x).
Complex roots
Always come in pairs
Non-real roots occur as conjugate pairs, so their count is even.

Your Results

Calculated
Positive Real Roots
-
Possible counts (sign changes in p(x))
Negative Real Roots
-
Possible counts (sign changes in p(-x))
Non-Real Complex Roots
-
Minimum count; rises in steps of 2
Polynomial Degree
-
Total roots, counted with multiplicity

Ready

Enter a polynomial in x and press Calculate.

How Descartes' Rule of Signs Works

Descartes' Rule of Signs is a quick algebraic test for bounding how many positive and negative real roots a polynomial p(x) with real coefficients can have, without solving the equation at all. It works entirely from the pattern of signs of the coefficients, written in standard form from the highest power of x down to the constant term.

Formula and method

Write the polynomial in standard form, powers of x in descending order, e.g. p(x) = aₙxⁿ + ... + a₁x + a₀. List only the nonzero coefficients in that order and count how many times consecutive signs differ — call this count V. The number of positive real roots (counted with multiplicity) equals V, or V minus an even number, down to 0 or 1. Next form p(-x) by substituting -x for x: this flips the sign of every coefficient attached to an odd power of x and leaves coefficients on even powers unchanged. Count the sign changes of the nonzero coefficients of p(-x) — call this W. The number of negative real roots equals W, or W minus an even number, down to 0 or 1. If the constant term is zero, x = 0 is also a root, with multiplicity equal to the lowest power of x that appears; the remaining roots split between the positive/negative real bounds above and non-real complex conjugate pairs, since a degree-n polynomial always has exactly n roots in total.

Common sources of error

  • Forgetting to flip odd-power signs: when forming p(-x), every term with an odd exponent changes sign (since (-x) raised to an odd power is negative), while even-power terms stay the same.
  • Counting through a missing term: a coefficient of zero is skipped entirely, not treated as a "sign" — compare only consecutive nonzero coefficients when counting changes.
  • Treating the count as exact: the number of sign changes is only an upper bound; the true number of positive (or negative) roots can be 2, 4, ... lower, never negative, and never below 0 or 1.
  • Ignoring multiplicity: Descartes' rule counts roots with multiplicity, so a repeated root (a double or triple root) is counted more than once in the tally.

Checking your result

Cross-check your sign-change counts against the Fundamental Theorem of Algebra: a degree-n polynomial has exactly n roots counted with multiplicity, split among positive real roots, negative real roots, a root at x = 0 (if the constant term is zero), and non-real complex roots that always come in conjugate pairs. If your positive count, negative count, zero multiplicity, and non-real count don't add up to n, or the non-real portion isn't even, recheck your sign tallies for p(x) and p(-x). For low-degree polynomials, you can also verify by factoring or graphing to confirm the actual number of real roots falls inside the predicted range.

Applications

Descartes' Rule of Signs is a fast first pass before committing to heavier root-finding work: it tells you how many positive and negative real roots to expect, which narrows the search interval for bisection, tells you how many starting guesses to try with Newton's method, and flags whether a characteristic equation from an engineering or control-theory model even has the real roots needed to be physically meaningful. It is usually taught alongside the Rational Root Theorem and synthetic division as a way to narrow down candidate roots before testing them by hand.

Frequently Asked Questions

What does Descartes' Rule of Signs tell you?
For a polynomial p(x) with real coefficients written in descending powers of x, the number of positive real roots equals the number of sign changes between consecutive nonzero coefficients of p(x), or is less than that by a multiple of 2, down to 0 or 1.
How do you find the bound on negative real roots?
Substitute -x for x to form p(-x); this flips the sign of every term with an odd power of x and leaves even-power terms unchanged. Count the sign changes between consecutive nonzero coefficients of p(-x) — the number of negative real roots equals that count, or is less by a multiple of 2.
Why can the actual root count be less than the number of sign changes?
Because non-real complex roots always occur in conjugate pairs. Each pair absorbs two of the roots implied by the sign changes, so the true positive (or negative) real root count can drop by 2, 4, 6, and so on below the sign-change count, but never below 0 or 1.
Does Descartes' Rule of Signs give the exact roots?
No. It only bounds how many positive, negative, and non-real roots a polynomial has — it does not compute their values. Use factoring, the quadratic formula, synthetic division, or a numerical method to solve for the actual roots.

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