The longest and shortest roots of a real cubic

Abstract

There are many formulas in the literature providing roots of a real cubic that avoid some of the well-known pathologies of Cardano's formulas. Among these, we identify two that consistently provide the unique roots of a depressed cubic that have the greatest and smallest absolute value, whenever those exist. We call these the longest and shortest roots. The existence conditions are elementary and are in terms of the signs of the coefficients and the discriminant. Our proofs use two algebraic identities satisfied by hypergeometric functions; once the standard real branches are fixed, the root comparisons are entirely real. As an application, the longest-root formula gives an explicit factorization of all but a vanishing proportion of depressed real quartics.