Polynomial roots and discriminants

The roots of polynomial equations in one variable can be shown easily to satisfy 2 independent linear partial differential equations with respect the equations coefficients. The discriminant also satisfies 2 similar independent LPDE’s.

Solving the quintic equation

A method is described for arriving at a power series expression for a real root of a Bring-Jerrard form quintic equation. The method relies on showing that any solution of the quintic must also satisfy a 4th order LDE. The 4th order LDE is easily solvable to give a power series expression for the solution.

Triangle inequalities.

An inequality for the area of a triangle A is derived in terms of the perimeter p. This inequality is related to the Weitzenbock inequality and is an improvement on it. The method used also throws up a second triangle inequality. A further link to the Hadwiger-Finsler inequality is also shown. The Ono inequality for acute angled triangles is also proved.

The CUBOID Inequalities

There are a number of inequalities that can be easily derived for the general cuboid. These inequalities connect the Perimeter P, Surface area S and the Volume V. The proof of one of these I have seen elsewhere, but I have not seen all of these published together. These can be easily applied to the general triangle via Heron’s formula. See the ‘Knol’ on triangle inequalities for the applications.

Solving cubic equations

A method is described for solving cubic equations. The roots of a cubic equation are treated as smooth functions of the equations’ coefficients and shown to be solutions to a second order linear differential equation which is easily solvable.