Descartes' Rule of Signs Calculator
Apply Descartes' Rule of Signs to determine the possible number of positive and negative real roots of a polynomial.
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Results
Sign changes (max positive real roots)
0
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- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Descartes' Rule of Signs Calculator
Descartes' Rule of Signs Calculator determines the possible number of positive and negative real roots of a polynomial by counting sign changes in the coefficient sequence. This elegant theorem, published by Rene Descartes in 1637, provides an upper bound on the number of positive real roots without actually solving the equation. The rule states that the number of positive real roots equals the number of sign changes in the coefficients or is less than that by an even number. To find the possible number of negative roots, substitute -x for x and count sign changes in the resulting polynomial. This technique is a valuable first step in root-finding, narrowing the search before applying the Rational Root Theorem or numerical methods.
The Math Behind It
Formula Reference
Descartes' Rule
# positive real roots = sign changes, or that minus an even number
Variables: Count sign changes in the coefficient sequence (skip zeros)
Negative roots
Evaluate P(-x) and count sign changes for negative root count
Variables: Replace x with -x, then apply the rule
Worked Examples
Example 1: Analyzing a cubic
Determine possible positive and negative roots of x³ - 4x² + x + 6
2 or 0 positive roots, 1 negative root
Common Mistakes & Tips
- !Counting sign changes incorrectly by including zero coefficients (skip them)
- !Forgetting that the number of roots can be the count MINUS any even number (not just the count itself)
- !Not applying the rule separately for positive (P(x)) and negative (P(-x)) roots
- !Confusing the rule with a precise count: it gives possible numbers, not the exact count
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
Does Descartes' Rule find the actual roots?
No. The rule only tells you how many positive and negative real roots are possible. It does not identify the roots themselves. Use the Rational Root Theorem, synthetic division, or numerical methods to find actual root values.
What about zero as a root?
Descartes' Rule does not count x = 0 as positive or negative. If a₀ = 0, factor out the highest power of x first (e.g., x³ + x² = x²(x+1)), apply the rule to the remaining polynomial, and add back the zero roots.
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