Factorizing Polynomials: Factorizing Polynomials from Two Rounds of Lin's Reduced Penultimate Remainder Process with a Real Divisor

IN a recent paper1 the author suggested that Lin's reduced penultimate remainder process could still be used to find the real or complex linear factors of polynomials of any degree, even when the process originally described by Lin2 was divergent, provided that the initial approximation to the linear factor being sought, and the two succeeding iterates, could be regarded as differing from the root sought by a small quantity of the first order. The main principle used was that in this linear case the convergence or divergence of the successive iterates to or from the root being sought was purely geometric, so that (for a sufficiently close starting approximation) the root being sought could be deduced from two iterations; complex numbers, however, might be involved in the iteration process.