Definition:Cubic Equation

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Definition

A cubic equation is a polynomial equation of the form:

$a x^3 + b x^2 + c x + d = 0$


Discriminant

Let:

$Q = \dfrac {3 a c - b^2} {9 a^2}$
$R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$

The discriminant of the cubic equation is given by:

$D := Q^3 + R^2$


Resolvent Equation

Let:

$y = x + \dfrac b {3 a}$
$Q = \dfrac {3 a c - b^2} {9 a^2}$
$R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$

Let $y = u + v$ where $u v = -Q$.


The resolvent equation of the cubic is given by:

$u^6 - 2 R u^3 - Q^3$


Also see

  • Results about cubic equations can be found here.


Historical Note

The ancient Greeks solved quadratics by geometric constructions, using the points of intersection of conic sections.

The general algebraic formulation of its solution remained unknown until as late as the end of the $15$th century, at which time Luca Bartolomeo de Pacioli wrote in his Summa de Arithmetica of $1494$ that the solution of the equations $x^3 + m x = n$ and $x^3 + n = m x$ were as impossible as Squaring the Circle.

However, this soon changed, as it was revealed shortly after that Scipione del Ferro was able to solve both of these, as well as the type $x^3 = m x + n$.

He passed this knowledge on to his apprentice Antonio Maria del Fiore.

Despite all attempts to keep the technique secret, it leaked out, and by $1535$ Niccolò Fontana Tartaglia also had the secret.

He was persuaded to tell Gerolamo Cardano, under an oath of secrecy.

However, Cardano learned that the secret had been obtained from Scipione del Ferro in the first place, so believed he was not bound by that oath, and so (to Tartaglia's annoyance) published it, in is Artis Magnae, Sive de Regulis Algebraicis of $1545$.


Sources

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