This is the method of choice for evaluating polynomials; it is faster and more numerically stable than the "normal" method, which involves computing the powers of x and multiplying them with the coefficients. The Horner scheme is often used to convert between different positional number systems (in which case x is the base of the number system, and the ai are the digits) and can also be used if x is a matrix, in which case the gain is even larger.