The Horner algorithm is important and useful for a variety of problems involving polynomial functions. On a practical level, it is the most efficient method for evaluation of polynomials, in the sense of requiring the fewest arithmetic operations. For use in a computer this feature not only saves time, but minimizes the opportunity for propagation of round‐off error. On a more formal level, the Horner algorithm turns out to be virtually equivalent to the procedure of synthetic division of polynomials, in the sense that the same steps used to perform one task also performs the other. This again has practical consequences. It allows us to find a quotient polynomial ‘for free’ in the process of evaluating a polynomial. Finally, since the Newton method for finding zeros of a polynomial involves nothing more complicated than finding the value of a polynomial at a point and then finding the value of the derivative of the polynomial at that point (which itself can be accomplished by another polynomial evaluation!), the Newton method itself can be programmed (to work for polynomials) by making suitable simple additions to a program for the Horner algorithm.