structural:scholar_edition, linguistic:translation
Polynomial root finding

Using the long division algorithm in combination with Newton's method, it is possible to approximate the real roots of a polynomial. The algorithm works as follows. Given a polynomial p n (x) {\displaystyle p_{n}(x)} of degree n {\displaystyle n} with zeros z n < z n − 1 < ⋯ < z 1, {\displaystyle z_{n}<z_{n-1}<\cdots <z_{1},} make some initial guess x 0 {\displaystyle x_{0}} such that z 1 < x 0 {\displaystyle z_{1}<x_{0}}. Now iterate the following two steps:

These two steps are repeated until all real zeros are found for the polynomial. If the approximated zeros are not precise enough, the obtained values can be used as initial guesses for Newton's method but using the full polynomial rather than the reduced polynomials.