论文标题
在带有siebeck的立方方程式上 - 梅尔赛 - 北胸三角形和四面体的四分之一方程式
On the Cubic Equation with its Siebeck--Marden--Northshield Triangle and the Quartic Equation with its Tetrahedron
论文作者
论文摘要
立方和四分之一多项式的真正根是在各自的siebeck-marden-northshield等边三角形和常规四面体的帮助下进行几何研究的。通过三角形的自由项的旋转来确定三立方根的根系的三角公式。提出了一个非常详细的$ x^4 + ax^3 + bx^2 + cx + d $的完整的根分类，以对单个系数$ a $ a $，$ b $，$ c $和$ d $强加条件。包含四分之一真实根的间隔的最大和最小长度由$ a $ a和$ b $确定。还发现了具有四个真实根的四分之一的上根和下根边界：没有根可以躺在$（\ sqrt {3}/4）\ sqrt {3a^2-8b} \，$中，$ a/a/4 $。四分之一的根源是通过在最多两个根中查找包含的间隔来定位的。这些间隔的终点取决于$ a $ a和$ b $，并且是二次方程式的根源 - 这使得此本地化对具有复杂参数系数的四分之一方程式有用。
The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck--Marden--Northshield equilateral triangle and regular tetrahedron. The Viète trigonometric formulae for the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classification for the quartic $x^4 + ax^3 + bx^2 + cx + d$ is proposed for which the conditions are imposed on the individual coefficients $a$, $b$, $c$, and $d$. The maximum and minimum lengths of the interval containing the four real roots of the quartic are determined in terms of $a$ and $b$. The upper and lower root bounds for a quartic with four real roots are also found: no root can lie farther than $(\sqrt{3}/4)\sqrt{3a^2 - 8b}\, $ from $-a/4$. The real roots of the quartic are localized by finding intervals containing at most two roots. The end-points of these intervals depend on $a$ and $b$ and are roots of quadratic equations -- which makes this localization helpful for quartic equations with complicated parametric coefficients.
