Math 2.0

Real functions Real polynomials and rational functions Theorem

Descartes' rule of signs

Builds on Polynomial ringRoot of a polynomialMultiplicity of a rootRolle's theoremBudan's theorem

Statement

Let f=anxn+⋯+a1x+a0f = a_n x^n + \cdots + a_1 x + a_0 be a nonzero polynomial with real coefficients. Let VV be the number of sign changes in the sequence of nonzero coefficients of ff, taken in order of degree. Let PP be the number of positive roots of ff, counted with multiplicity. Then P≤VP \le V, and V−PV - P is even.

Remarks

The parity part follows by comparing the sign of the leading coefficient with the sign of the lowest nonzero coefficient. Both VV and PP are even exactly when these signs agree. The bound P≤VP \le V can be proved by induction on the degree, using Rolle's theorem to compare the positive roots of ff with those of f′f'. Example: f=x3−x2−x+1f = x^3 - x^2 - x + 1 has signs +,−,−,++, -, -, +, so V=2V = 2; indeed f=(x−1)2(x+1)f = (x - 1)^2 (x + 1) has the double positive root 11, so P=2P = 2. The number of negative roots is bounded in the same way by the sign changes of f(−x)f(-x). The bound need not be reached: x2−x+1x^2 - x + 1 has V=2V = 2 and no real root, so P=0P = 0.

Going deeper

For a student

The rule is a quick first test before one locates roots more precisely. If all roots of ff are real, then the bound is exact: P=VP = V. Budan's theorem extends the rule from the half-line (0,∞)(0, \infty) to any interval, and Sturm's theorem gives the exact number of distinct real roots in an interval.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPolynomial greatest common divisor, concept, no evidence levelPolynomial greatest c…conceptPolynomial long division, concept, no evidence levelPolynomial long divis…conceptSequence, definition, no evidence levelSequencedefinitionRing, definition, no evidence levelRingdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionIntermediate value theorem, statement, no evidence levelIntermediate value th…statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementMultiplicity of a root, definition, no evidence levelMultiplicity of a rootdefinitionSturm's theorem, statement, no evidence levelSturm's theoremstatementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementBudan's theorem, statement, no evidence levelBudan's theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementDescartes' rule of signs, statement, no evidence levelDescartes' rule of si…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Limit point of a set definition
  3. Local maximum definition
  4. Polynomial greatest common divisor concept
  5. Polynomial long division concept
  6. Sequence definition
  7. Ring definition
  8. Limit of a function definition
  9. Local minimum definition
  10. Real sequence definition
  11. Subsequence definition
  12. Polynomial ring definition
  13. Continuity at a point definition
  14. Derivative definition
  15. Bounded sequence definition
  16. Limit of a sequence definition
  17. Monotone sequence definition
  18. Degree of a polynomial definition
  19. Polynomial evaluation definition
  20. Continuous function definition
  21. Fermat's theorem on stationary points statement
  22. Convergent sequence definition
  23. Monotone convergence theorem for sequences statement
  24. Every sequence has a monotone subsequence statement
  25. Root of a polynomial definition
  26. Intermediate value theorem statement
  27. Bolzano–Weierstrass theorem statement
  28. Multiplicity of a root definition
  29. Sturm's theorem statement
  30. Boundedness theorem statement
  31. Budan's theorem statement
  32. Extreme value theorem statement
  33. Rolle's theorem statement
  34. Descartes' rule of signs statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Descartes%27s_rule_of_signs
About this page
TagT6DQQ36JZ5
URIm2://T6DQQ36JZ5
Kindstatement, theorem
WritingW0 stub
MSC26C10, 12D10
StatusThis copy of the site has no build data. Run m2 status T6DQQ36JZ5.
AboutRoot of a polynomial
UsesPolynomial ring, Root of a polynomial, Multiplicity of a root, Rolle's theorem, Budan's theorem
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Changed2 November 2026

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