Difference between revisions of "User:Temperal/The Problem Solver's Resource4"
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*<math>rs+st+rt=\frac{-c}{b}</math> | *<math>rs+st+rt=\frac{-c}{b}</math> | ||
<!-- actually, I'm not sure if this is right. Could someone check this? --> | <!-- actually, I'm not sure if this is right. Could someone check this? --> | ||
+ | |||
+ | The three roots <math>r, s, t</math> of a cubic polynomial equation <math>x^3 + ax^2 + bx + c = 0</math> are given implicitly by | ||
+ | <math>r</math> <math>=</math> <math>- \frac {a}{3} + \left(\frac { - 2a^3 + 9ab - 27c + \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}</math> | ||
+ | <math>{} + \left(\frac { - 2a^3 + 9ab - 27c - \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}</math> | ||
+ | <math>s</math> <math>=</math> <math>- \frac {a}{3} - \frac {1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c + \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}</math> | ||
+ | <math>{} + \frac { - 1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c - \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}</math> | ||
+ | <math>t</math> <math>=</math> <math>- \frac {a}{3} + \frac { - 1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c + \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}</math> | ||
+ | <math>{} - \frac {1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c - \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}</math> | ||
Quartic formulas are listed [http://www.josechu.com/ecuaciones_polinomicas/cuartica_solucion.htm here]. | Quartic formulas are listed [http://www.josechu.com/ecuaciones_polinomicas/cuartica_solucion.htm here]. |
Revision as of 17:22, 8 December 2007
AlgebraThis is a collection of algebra laws and definitions. Obviously, there is WAY too much to cover here, but we'll try to give a good overview. Elementary AlgebraDefinitions
Factor TheoremIff a polynomial Quadratic FormulaFor a quadratic of form Fundamental Theorems of Algebra
Third-degree and Quartic FormulasIf third-degree polynomial The three roots
Quartic formulas are listed here. The general quintic equation (or an equation of even higher degree) does not have a formula. DeterminantsThe determinant of a General Formula for the DeterminantLet
Cramer's LawConsider a set of three linear equations (i.e. polynomials of degree one) Let Abstract AlgebraIncomplete. Diophantine EquationsIncomplete. |