Editing Expression (mathematics) (section)

=== Polynomial evaluation ===
{{Main|Polynomial evaluation}}
A polynomial consists of variables and [[coefficient]]s, that involve only the operations of [[addition]], [[subtraction]], [[multiplication]] and [[exponentiation]] to [[nonnegative integer]] powers, and has a finite number of terms. The problem of [[polynomial evaluation]] arises frequently in practice. In [[computational geometry]], polynomials are used to compute function approximations using [[Taylor polynomials]]. In [[cryptography]] and [[hash table]]s, polynomials are used to compute [[K-independent hashing|''k''-independent hashing]].

In the former case, polynomials are evaluated using [[floating-point arithmetic]], which is not exact. Thus different schemes for the evaluation will, in general, give slightly different answers. In the latter case, the polynomials are usually evaluated in a [[finite field]], in which case the answers are always exact.

For evaluating the [[univariate polynomial]] <math display="inline">a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0,</math> the most naive method would use <math>n</math> multiplications to compute <math>a_nx^n</math>, use <math display="inline">n-1</math> multiplications to compute <math>a_{n-1} x^{n-1}</math> and so on for a total of <math display="inline">\frac{n(n+1)}{2}</math> multiplications and <math>n</math> additions. Using better methods, such as [[Horner's rule]], this can be reduced to <math>n</math> multiplications and <math>n</math> additions. If some preprocessing is allowed, even more savings are possible.