In Search for Radical Roots of Polynomial Equations of degree n > 1

Math Online Tom Circle

Take note: Find roots 根 to solve polynomial 多项式方程式equations, but find solutionto solve algebraic equations代数方程式.

Radical : (LatinRadix = root): $latex sqrt [n]{x} $

Quadratic equation (二次方程式) 有 “根式” 解:[最早发现者 : Babylon 和 三国时期的吴国 数学家 赵爽]

$latex {a.x^{2} + b.x + c = 0}&fg=aa0000&s=3$

$latex boxed{x= frac{-b pm sqrt{b^{2}-4ac} }{2a}}&fg=aa0000$

Cubic Equation: 16 CE Italians del Ferro, Tartaglia & Cardano
$latex {a.x^{3} = p.x + q }&fg=0000aa&s=3$

Cardano Formula (1545 《Ars Magna》):
$latex boxed {x = sqrt [3]{frac {q}{2} + sqrt{{ (frac {q}{2})}^{2} – { (frac {p}{3})}^{3}}}
+ sqrt [3]{frac {q}{2} -sqrt{ { (frac {q}{2})}^{2} – { (frac {p}{3})}^{3}}}}&fg=0000aa$

Quartic Equation: by Cardano’s student Ferrari
$latex {a.x^{4} + b.x^{3} + c.x^{2} + d.x + e = 0}&fg=00aa00&s=3$

Quintic Equation:
$latex {a.x^{5} + b.x^{4} + c.x^{3} + d.x^{2} + e.x + f = 0}&s=3$

No radical solution (Unsolvability) was suspected by Ruffini (1799)…

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