A polynomial A(x) = a₀ + a₁x + a₂x² + a₃x³ + ... may be written as A(x) = a₀ + x(a₁ + x(a₂ + x(a₃ + ...))).

A polynomial may be evaluated at a point x', that is A(x') computed, in Θ(n) time using Horner's rule. That is, repeated multiplications and additions, rather than the naive methods of raising x to powers, multiplying by the coefficient, and accumulating.