Math July 13, 2026 · 8 Min Read

Descartes' Rule of Signs Calculator – Guide & Formulas

Apply Descartes' Rule of Signs to find the possible number of positive and negative real roots of any polynomial. Step-by-step sign change analysis.

Apply Descartes' Rule of Signs to any polynomial instantly with our free calculator. Enter coefficients to find the possible number of positive and negative real roots with step-by-step sign analysis.

Key Takeaway

Use the free Descartes' Rule of Signs Calculator to apply descartes' rule of signs to find the possible number of positive and negative real roots of any polynomial. step-by-step sign change analysis. Get instant results with step-by-step explanations.

How to Use the Descartes' Rule of Signs Calculator

  1. Enter polynomial coefficients from highest degree to lowest, separated by commas.
  2. Use 0 for missing terms (e.g., x³ - 2x + 1 → 1, 0, -2, 1).
  3. Review the number of sign changes in f(x) for positive roots.
  4. Review the number of sign changes in f(-x) for negative roots and all valid combinations.

The Formula

Count sign changes between consecutive non-zero coefficients of f(x) to get V+ (max positive roots). Substitute -x to get f(-x), count sign changes to get V- (max negative roots). Possible counts are V+, V+-2, V+-4, ... down to 0 or 1.

Variable Definitions

  • V+: Number of sign changes in f(x), giving the maximum possible positive real roots
  • V-: Number of sign changes in f(-x), giving the maximum possible negative real roots
  • f(x): The polynomial with real coefficients ordered by descending degree
  • f(-x): The polynomial obtained by substituting -x for x (negates odd-degree coefficients)
  • n: Degree of the polynomial (highest power)

Applying Descartes' Rule to x³ - 3x + 2

Find the possible number of positive and negative real roots.

  1. Step 1: Coefficients of f(x) = x³ + 0x² - 3x + 2 are: 1, 0, -3, 2. Non-zero: 1, -3, 2.
  2. Step 2: Sign changes in f(x): +1→-3 (1 change), -3→+2 (2 changes). V+ = 2.
  3. Step 3: f(-x) = -x³ + 3x + 2. Coefficients: -1, 3, 2. Sign changes: -1→+3 (1 change). V- = 1.
  4. Step 4: Possible combinations: (2 positive, 1 negative, 0 complex) or (0 positive, 1 negative, 1 complex pair).

Frequently Asked Questions

What is Descartes' Rule of Signs?

Descartes' Rule of Signs is a classical algebra theorem that determines the possible number of positive and negative real roots of a polynomial by counting sign changes in its coefficients.

How do I enter polynomial coefficients?

Enter coefficients from highest degree to lowest (constant term), separated by commas or spaces. Use 0 for missing terms. For x³ - 2x + 1, enter: 1, 0, -2, 1.

Does Descartes' Rule give the exact number of roots?

No, it gives an upper bound. The actual number of positive (or negative) real roots is either equal to the number of sign changes or less by an even number. For example, 3 sign changes means 3 or 1 positive real roots.

What about zero roots?

Descartes' Rule does not count zero as a root. Factor out x as many times as possible (check if the constant term is zero), then apply the rule to the remaining polynomial.

Why do complex roots come in pairs?

For polynomials with real coefficients, complex roots always come in conjugate pairs (a + bi and a - bi). This is why the difference between sign changes and actual roots is always even.

What is f(-x)?

To find f(-x), replace every x with -x in the polynomial. This negates all odd-degree coefficients and keeps even-degree coefficients unchanged. Counting sign changes in f(-x) gives the maximum number of negative roots.

Can a polynomial have no sign changes?

Yes. If there are zero sign changes in both f(x) and f(-x), the polynomial has no positive and no negative real roots (all roots are complex, except possibly zero).

How is this different from the quadratic formula?

Descartes' Rule only tells you the possible number of roots, not their values. The quadratic formula (for degree 2), cubic formula (degree 3), or numerical methods are needed to find actual root values.