Estimate positive, negative, zero, and non-real roots using Descartes’ Rule of Signs. Enter coefficients individually as a₀, a₁, a₂ ... aₙ corresponding to powers of x from lowest to highest. Example: a₀ = 1, a₁ = -6, a₂ = 11, a₃ = -6 for p(x) = -6x³ + 11x² - 6x + 1.
p(x) =
This calculator estimates the possible number of positive, negative, zero, and non-real roots of a polynomial using Descartes' Rule of Signs. Simply enter the coefficients of your polynomial, and the calculator instantly analyzes sign changes to determine possible root counts.
Enter coefficients individually as:
The polynomial formula updates automatically as you type.
The calculator:
Results are displayed as possible values (e.g., 3, 1).
If the polynomial begins with zero constant terms:
This ensures accurate handling of polynomials like x²(x - 3).
As you modify coefficients:
1️⃣ Enter coefficients from lowest power (a₀) to highest power (aₙ).
2️⃣ The calculator removes leading zero coefficients (highest degree zeros).
3️⃣ It counts sign changes in:
p(x) → gives possible positive roots.
p(-x) → gives possible negative roots.
4️⃣ The number of positive (or negative) real roots equals the number of sign changes
or less than that by an even number.
5️⃣ Zero roots are detected from repeated zero constant terms.
6️⃣ Any remaining roots (based on polynomial degree) are counted as non-real complex roots.
This method does not compute exact roots — it only estimates the possible number of real roots using sign analysis.
Descartes’ Rule of Signs is a mathematical rule used to estimate the possible number of positive and negative real roots of a polynomial equation. Instead of solving the polynomial directly, the rule analyzes the pattern of positive and negative coefficients in the polynomial.
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Using this polynomial form, Descartes’ Rule of Signs works by counting how many times the sign of the coefficients changes when moving from the highest power of x to the constant term.
For example, if a polynomial has three sign changes, it can have 3 or 1 positive real roots. The rule provides possible counts, not the exact root values. To find exact solutions, additional methods such as factoring or numerical algorithms must be used.
This rule is widely used in algebra to quickly analyze the structure of polynomial equations and determine how many real solutions may exist before attempting more complex calculations.
Consider the polynomial:
p(x) = x³ - 6x² + 11x - 6
Sign changes in p(x): 3 → Possible positive roots: 3 or 1
Sign changes in p(-x): 0 → Possible negative roots: 0
Since this is a degree 3 polynomial, the total number of real roots cannot exceed 3. Any remaining roots would be non-real complex roots.
Learn how this calculator estimates possible positive, negative, zero, and non-real roots of a polynomial.
This calculator applies Descartes' Rule of Signs to estimate the possible number of positive real roots, negative real roots, zero roots (multiplicity), and minimum non-real roots of a polynomial based on the coefficients you enter.
Enter coefficients individually starting from a₀ (constant term) up to aₙ (highest power). The coefficients correspond to powers of x from lowest to highest. You can click “Add Coefficient” to insert higher-degree terms.
The calculator counts the number of sign changes in the polynomial’s coefficients (after removing leading zeros). The possible number of positive real roots equals that number or less by an even integer.
The calculator substitutes −x into the polynomial by alternating the signs of odd-powered coefficients. It then counts sign changes to estimate the possible number of negative real roots.
If the constant term (a₀) is zero, the calculator removes it and counts how many times this occurs. Each zero constant term increases the multiplicity of x = 0 as a root.
After calculating the maximum possible real roots (positive, negative, and zero), the calculator subtracts that total from the polynomial’s degree. The remaining roots must be non-real complex roots.
No. Descartes' Rule of Signs only provides the possible number of positive and negative real roots. It does not compute the exact root values.
If any coefficient field is empty or contains invalid data, the calculator displays an error message and hides the result until valid numbers are entered.
Yes. You can continue adding coefficients to increase the degree. The calculator dynamically updates and evaluates sign changes regardless of polynomial degree.
Yes. Use the “Clear all” button to reset inputs or the “Reload calculator” button to restore the default state instantly.
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