Descartes’ Sign Rule

Summary: The number of positive roots of a polynomial equals the number of sign changes between consecutive nonzero coefficients; the number of negative roots equals the number of consecutive same-sign pairs (successions).

Sources: chapter-1.4.11, chapter-1.4.13

Last updated: 2026-05-03

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Statement

For a polynomial equation written with all terms on one side, arranged by descending degree, and with nonzero integer coefficients:

• Positive roots: count the number of times the sign changes from one term to the next (+ to − or − to +).
• Negative roots: count the number of times the same sign is repeated consecutively (successions).

The number of positive (resp. negative) real roots is at most equal to the count and has the same parity (they may be fewer by 2, 4, … because complex roots come in conjugate pairs). Euler states the rule without the parity caveat, treating the counts as exact (source: chapter-1.4.11, §725; chapter-1.4.13, §758).

Example: cubic (§725)

For $x^3-a{x}^2+bx-c=0$ with $a,b,c>0$:

$+,-,+,-\Rightarrow \text{3 sign changes}\Rightarrow \text{3 positive roots}$

For $x^3+a{x}^2+bx-c=0$:

$+,+,+,-\Rightarrow \text{1 sign change, 2 successions}\Rightarrow \text{1 positive root, 2 negative roots}$

Example: quartic (§758–759)

$y^4-3y^3+12y^2-162y+72=0$: signs $+,-,+,-,+$ → 4 changes, 0 successions → all four roots positive; no need to test negative divisors.

$x^4+2x^3-7x^2-8x+12=0$: signs $+,+,-,-,+$ → 2 changes, 2 successions → 2 positive, 2 negative roots. Confirmed: roots $1,2,-3,-4$.

Practical use

The rule narrows which divisors of the constant term to test when applying the rational-root-theorem: only positive divisors need be tried if the rule predicts no negative roots, and vice versa.

Related pages

• rational-root-theorem
• quartic-equations
• cubic-equations
• ch1.4.11-complete-cubic-equations
• ch1.4.13-quartic-equations
• equations