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POLYNOMIALS

DOI: 10.1615/AtoZ.p.polynomials

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.

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