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An expression of the form

where at least an ≠ 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 xi; for i = 1, 2, ... , n. Therefore, since Pn(x) is equal to zero for each x = xi, it can be written as

the last expression is called the factorization of Pn(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 − a1/a0.

From the Fundamental Theorem of Algebra it also follows that the representation of a polynomial Pn(x) in terms of powers of x is unique. That is, a polynomial Pn(x) is uniquely associated with a set of n + 1 coefficients ai 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.

### Calculation of Numerical Values of a Polynomial

Notice that the calculation for a given value of the argument x of a quadratic polynomial P2(x) = a0 + a1x + a2x2 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 P2(x) this is: P2(x) = (a2x + a1)x + a0, 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.

### Other Types of Polynomials

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, x2, … , xn}, 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.

#### Les références

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