Historically, calculus has displaced many algebraic methods for solving classical problems. This article illustrates an algebraic method for finding the zeros of polynomial functions that is closely related to Newton's method (devised in 1669, published in 1711), which is encountered in calculus. By exploring this problem, precalculus students are introduced to the notable mathematicians Newton, Raphson, Lagrange, and Euler--mathematicians whose work is usually beyond the scope of precalculus courses in school mathematics. With the aid of CAS, students can explore some of the same problems that these historical figures did, thereby combining the history of mathematics with modern tools for analyzing functions. (Contains 8 figures and 1 table.)