Descartes' Rule of Signs in 2026 Polynomial Algebra
Formulated in 1637 by French philosopher and mathematician René Descartes, Descartes' Rule of Signs provides an elegant method for determining the maximum number of positive and negative real roots of a polynomial without factoring or graphing.
Core Rules Reference Table
| Evaluated Form | Counted Metric | Permissible Real Roots |
|---|---|---|
| $P(x)$ | Number of sign variations ($v$) | Equal to $v$, or less by an even integer ($v - 2k$) |
| $P(-x)$ | Number of sign variations ($v'$) | Equal to $v'$, or less by an even integer ($v' - 2k$) |
| Complex Conjugates | Pairs of roots ($2k$) | Accounts for missing real roots ($n - N_{\text{real}}$) |