An expression of the form

where at least a_{n} ≠ 0, is called a polynomial of degree n. The * Fundamental Theorem of Algebra* states that a polynomial of degree n has exactly n roots or zeros, which can be real or complex. Let us call them x_{i}; for i = 1, 2, ... , n. Therefore, since P_{n}(x) is equal to zero for each x = x_{i}, it can be written as

the last expression is called the factorization of P_{n}(x).

Relations between the roots and the coefficients of a polynomial follow immediately expanding this last expression. For example, the sum of the roots is equal to − a_{1}/a_{0}.

From the Fundamental Theorem of Algebra it also follows that the representation of a polynomial P_{n}(x) in terms of powers of x is unique. That is, a polynomial P_{n}(x) is uniquely associated with a set of n + 1 coefficients a_{i} for i = 0, 1, 2, … , n. Polynomials of low degree receive special names: a linear polynomial is one of the first degree; quadratic is one of the second degree, cubic is one of the third degree.

Notice that the calculation for a given value of the argument x of a quadratic polynomial P_{2}(x) = a_{0} + a_{1}x + a_{2}x^{2} requires two additions and three multiplications, one for the second term and two for the third. In general to calculate a value of a polynomial of degree n it is necessary to do n additions and n(n + l)/2 multiplications. A considerable saving in computing time can be achieved using the so-called nested form of a polynomial. For P_{2}(x) this is: P_{2}(x) = (a_{2}x + a_{1})x + a_{0}, the number of additions necessary to calculate a value of it for a given value of x remains equal to n = 2 but the number of multiplications is reduced from 3 to 2.

In general, for a polynomial of degree n, the nested form has the general expression

using the nested form the number of multiplications is reduced substantially: from n(n + 1)/2 to simply n.

Up to now we have referred only to polynomials in terms of powers of x, that is, we have used the basis of representation {1, x, x^{2}, … , x^{n}}, but instead of that basis of representation we could use, for example, the trigonometric functions:

and have then the trigonometric polynomial

Finite sections of Fourier Series are a special type of *trigonometric polynomials.* Similarly, we can define polynomials in terms of other representation basis: for example, Legendre or Chebyshev Polynomials of degrees 0, 1, 2, … , Bessel functions of orders 0, 1, 2, …, and many others. Such representations are useful in applications, for example, for the solution of Differential Equations.

#### REFERENCES

Fike, C. T. (1968) *Computer Evaluation of Mathematical Functions*, Prentice-Hall, New Jersey.