Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solving cubic equations by completing the cube and higher degree equations by completing powers

Hua-Lin Huang, Shengyuan Ruan, Xiaodan Xu, Yu Ye

Abstract

We derive the Cardano formula of cubic equations by completing the cube, and provide radical solutions to some algebraic equations of higher degree by completing powers. The main idea of completing powers arises from Harrison's center theory of higher degree forms. A very simple criterion for such algebraic equations is presented, and the computation amounts to solving linear equations and quadratic equations.

Solving cubic equations by completing the cube and higher degree equations by completing powers

Abstract

We derive the Cardano formula of cubic equations by completing the cube, and provide radical solutions to some algebraic equations of higher degree by completing powers. The main idea of completing powers arises from Harrison's center theory of higher degree forms. A very simple criterion for such algebraic equations is presented, and the computation amounts to solving linear equations and quadratic equations.
Paper Structure (5 sections, 8 theorems, 48 equations)

This paper contains 5 sections, 8 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Harrison centers and completing powers
  3. Cardano formula revisited by completing the cube
  4. Solving some algebraic equations by completing powers
  5. Summary

Key Result

Proposition 2.2

Suppose $f \in \mathbbm{k}[x_1, x_2, \dots, x_n]$ is a nondegenerate homogeneous polynomial of degree $d>2.$ Then $Z(f)$ is a commutative algebra, and $f$ is diagonalizable over $\mathbbm{k}$ if and only if $Z(f) \cong \mathbbm{k}^n$ as algebras.

Theorems & Definitions (23)

  • Example 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • ...and 13 more