Since no general formulae exist to compute the zeros of a high order polynomial f(x), Matlab uses sophisticated "guess and check" algorithms to numerically approxmate the zeros. One common such algorithm is known as Newton's method, which can be summarized as follows: Start with an (essentially arbitrary) initial guess z for one of the zeros of the polynomial f(x). Compute f(z). If f(z)=0 then z is a zero (root) of the polynomial and we are done. Of course, it is unlikely that the initial guess for z really is a zero, so generally f(z)=0. In this case, Newton's method generates a new, "improved" guess for z according to the formula z=zf(z)f(z) In this formula the "old" guess for z is used on the right-hand side of the equation, and the result is used to define the new guess for z. We then check this new z to see if f(z)=0, and if not repeat the above calculation to generate a new z, continuing as long as necessary until f(z)=0. Of course, the finite precision of the computer means that we can't really continue the process of generating better and better guesses until f(z)=0, as this would require essentially infinite time. Rather, must settle for weros which satify f(z)