Descartes' rule of signs

Observation — Descartes' rule of signs

In the sequence of signs of the coefficients $a,b,c$:

• every persistence of sign (e.g. $+,+$) corresponds, if it exists, to a negative solution;
• every variation of sign (e.g. $+,-$) corresponds, if it exists, to a positive solution.

Example: $2x^2+8x-32=0$. Signs: $+,+,-$ (persistence then variation). Hence $|\text{neg. sol.}|>\text{pos. sol.}$

The rule follows directly from Vieta’s formulas: the sign of the sum and of the product of the roots depend on the signs of the coefficients, and from these one recovers the sign of each solution.

Links

Topics: Second-degree equations
Concepts: Rule of signs · Sum and product of the roots
Skills: Reasoning by cases
People: René Descartes (Cartesio)