Math CentralQuandaries & Queries


Question from Evin, a student:

hello.i am a student . ax^4+bx^3+cx^2+dx+e=0 x=? i want to learn the solution or formula of equations of the fourth and fifth degree ...PLEASE


My first reaction is "oh, no, you don't want to" <grin> But since you asked... unto Caesar thou shalt go.

For fourth degree equations you use the "quartic formula." This is truly horrible as a single formula, but somewhat less so as a sequence of smaller steps. See for instance

There is, and can be, no general formula for quintics using only rational functions and roots (this was proved by Abel and Galois.) The reason is that the symmetry group of the solutions of a general quintic is S5 [the full permutation group on five objects], which cannot be built up from the cyclic groups that give the symmetries of radicals. It's like a Rubik's Cube with not enough bits movable.

Obviously some quintics (eg, x5 - 32 = 0) can be solved.

General quintics can be solved exactly using theta functions (compare, if you are familiar with it, the technique for solving cubics using trig functions).

Good Hunting!


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