The convergence is generally quadratic, it may converge much slowly or even not converge at all. In particular, if the polynomial has no real root, and formula_7 is real, then Newton's method cannot converge. However, if the polynomial has a real root, which is larger than the larger real root of its derivative, then Newton's method converges quadratically to this largest root if formula_7 is larger that this larger root (there are easy ways for computing an upper bound of the roots, see Properties of polynomial roots). This is the starting point of Horner method for computing the roots.