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Descartes' Rule of Signs Calculator
Apply Descartes' Rule of Signs to any polynomial. Count sign changes in f(x) and f(−x) to determine the possible number of positive, negative, and complex roots.
Enter coefficients from highest to lowest power, separated by commas
Polynomial f(x)⧉
x^3 −6x^2 +11x −6
Degree 3 polynomial
Sign changes in f(x)⧉
3
Possible positive real roots: 3, 1
f(−x)⧉
−x^3 −6x^2 −11x −6
Polynomial with x replaced by −x
Sign changes in f(−x)⧉
0
Possible negative real roots: 0
Degree⧉
3
Total number of roots (real + complex) counting multiplicity
Root Combinations⧉
2
Number of distinct possible root-type distributions
Sign Changes in f(x)
| Term | Coefficient | Sign | Change? |
|---|---|---|---|
| x^3 | 1 | + | — |
| x^2 | -6 | − | ⇄ YES |
| x^1 | 11 | + | ⇄ YES |
| x^0 | -6 | − | ⇄ YES |
Sign Changes in f(−x)
| Term | Coefficient | Sign | Change? |
|---|---|---|---|
| x^3 | -1 | − | — |
| x^2 | -6 | − | — |
| x^1 | -11 | − | — |
| x^0 | -6 | − | — |
Possible Root Distributions
| Positive Real | Negative Real | Complex | Total |
|---|---|---|---|
| 3 | 0 | 0 | 3 |
| 1 | 0 | 2 | 3 |
Visual Summary
Option 1: 3 positive, 0 negative, 0 complex
+3
Option 2: 1 positive, 0 negative, 2 complex
+1
2i
Planning notes, formulas, and examples
About the Descartes' Rule of Signs Calculator
Descartes' Rule of Signs is a powerful theorem in algebra that tells you the possible number of positive and negative real roots of a polynomial without actually solving it. The rule states that the number of positive real roots of a polynomial f(x) is either equal to the number of sign changes in the sequence of its coefficients, or less than that by an even number. Similarly, the number of negative real roots equals the number of sign changes in f(−x), or less by an even number. Any remaining roots must be complex.
This calculator lets you enter the coefficients of any polynomial and see the sign-change analysis for both f(x) and f(−x). It highlights every sign change in the coefficient sequence, lists all possible distributions of positive, negative, and complex roots, and presents the results in a visual stacked-bar format. Two input modes are available: comma-separated coefficients for quick entry, or individual fields for each degree. Presets for classic polynomials let you explore the rule immediately. Whether you are narrowing down root possibilities before applying the Rational Root Theorem, checking your factoring work, or studying polynomial behavior in an algebra or precalculus course, Descartes' Rule gives you a fast, reliable upper bound on the number of roots of each type.
When This Page Helps
descartes-rule-of-signs helps you solve descartes-rule-of-signs problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Coefficients (highest degree first), constant once and immediately inspect Polynomial f(x), Sign changes in f(x), f(−x) to validate your work.
How to Use the Inputs
- Enter Coefficients (highest degree first) and constant in the input fields.
- Select the mode, method, or precision options that match your descartes-rule-of-signs problem.
- Read Polynomial f(x) first, then use Sign changes in f(x) to confirm your setup is correct.
- Try a preset such as "x³−6x²+11x−6" to test a known case quickly.
Formula used
Descartes' Rule: The number of positive real roots of f(x) equals the number of sign changes in the coefficient sequence, or less by a positive even integer. For negative roots, apply the same rule to f(−x). Complex roots account for the remainder.Example Calculation
Result: Polynomial f(x) shown by the calculator
Using the preset "x³−6x²+11x−6", the calculator evaluates the descartes-rule-of-signs setup, applies the selected algebra rules, and reports Polynomial f(x) with supporting checks so you can verify each transformation.
Tips & Best Practices
- Descartes' Rule gives an upper bound — the actual count may be less by an even number.
- Zero coefficients are skipped when counting sign changes.
- To find the sign changes in f(−x), note that odd-degree coefficients flip sign.
- Combine with the Rational Root Theorem to narrow down candidate roots after applying Descartes' Rule.
- Complex roots always come in conjugate pairs for polynomials with real coefficients — so the complex count is always even.
How This descartes-rule-of-signs Works
This calculator takes Coefficients (highest degree first), constant and applies the relevant descartes-rule-of-signs relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Interpreting Results
Start with the primary output, then use Polynomial f(x), Sign changes in f(x), f(−x), Sign changes in f(−x) to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
Study Strategy
Sources & Methodology
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Frequently Asked Questions
It tells you the maximum possible number of positive and negative real roots of a polynomial, and that the actual count differs from the maximum by an even number. Any remaining roots are complex.
No. It gives a list of possibilities. For example, 3 sign changes means 3 or 1 positive roots. You need other methods (Rational Root Theorem, graphing, or numerical solving) to determine the exact count.
Zero coefficients are skipped. A sign change is counted only between consecutive nonzero coefficients.
Because complex roots of a real polynomial come in conjugate pairs, removing 2 real roots at a time and replacing them with 2 complex roots. So the count drops by 2, 4, etc.
No. It only counts possible numbers of positive and negative roots. To find the actual root values, use factoring, the Rational Root Theorem, synthetic division, or numerical methods.
No. Descartes' Rule of Signs applies only to polynomials with real coefficients. The notion of 'sign' is not defined for complex numbers.
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