4 ms·

True -- I got overzealous with x^5, some (not all) can be factored. I'll reword to "some more complex systems.

• psykotic 14y ago
  > I got overzealous with x^5, some (not all) can be factored All polynomials can be factored but only some of the fifth or higher degree are solvable by radicals--a crucial distinction. Here's a fun puzzle that also serves as a gentle introduction to the central idea of Galois theory: Show that any palindromic quintic is solvable by radicals. A palindromic quintic is one of the form ax^5 + bx^4 + cx^3 + cx^2 + bx + a. It might be useful to know that if you have a polynomial equation f(z) = 0 and make the substitution z -> 1/z and multiply through by z^n, you get the reversed polynomial. On the Riemann sphere, this inversion symmetry reflects the northern hemisphere onto the southern hemisphere and vice versa, interchanging the north and south pole, which are 0 and infinity in z coordinates. Therefore a palindromic polynomial is one that "looks the same" from either vantage point. This idea of looking at the behavior of a polynomial simultaneously from the vantage point of 0 and infinity is also the basis of how the fundamental theorem of algebra is proved. The general model is 1 + z^n. Once you understand that polynomial qualitatively near 0 and infinity, you can understand every other polynomial by a simple perturbation analysis.