Although a closed form solution exists for the roots of polynomials of degree ≤ 4, the general formulae for cubics (and quartics) is ugly. Various simplifications can be made; commonly, the cubic \(a_3\,x^3+a_2\,x^2+a_1\,x+a_0\) is transformed by substituting \(x = t-a_2/3a_3\), giving Continue reading Solving Cubic Polynomials
A well known (if not by name) theorem is the Abel–Ruffini theorem, which states that there is no algebraic expression for the roots of polynomials with degree higher than 4.
A not-so-well-known fact is that for any polynomial \(P(x)\), it is possible to find (with exact arithmetic) a set of ranges each containing exactly one root of \(P(x)\). One such algorithm is due to James Victor Uspensky in 1948. Continue reading Solving Polynomials
Hell is freezing over. Or something. This month I've created a twitter and now a blog-type thing here. Welcome to 2006, I know.