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