Polynomials
Polynomials in one variable x are algebraic expressions of the form
P(x) = c0 + c1x + c2x2 + c3x3 + ... + cNxN.
Here c0, c1, c2, ... cN are called the coefficients of the polynomial. If cN 0 then the degree of this polynomial is the nonnegative integer N. A polynomial in one variable can also be conveniently expressed using the summation notation as
Then a polynomial in two variables x and y is an algebraic expression of the form
Polynomials in three or more variables are defined in a similar manner. In general the coefficients of a polynomial can belong to an arbitrary commutative ring. But in this article we will restrict our attention to polynomials in one variable and with coefficients in the field of complex numbers C, or in a familiar subring of this field. For more information on polynomials in two or more variables, see the article on algebraic geometry.
Let C[x] denote the set of polynomials in one variable x and having coefficients in the field C of complex numbers. We define addition of polynomials in this set by adding the corresponding coefficients, so that
Here N = max{L,M}, and it must be understood that al = 0 if l > L and bm = 0 if m > M. The definition of multiplication is slightly more complicated. We set
where K = L + M, and for each integer k with 0 k K we define
ck = a0bk + a1bk-1 + a2bk-2 + ... + ak-1b1 + akb0.
Again it must be understood that al = 0 if l > L and bm = 0 if m > M. With these definitions of addition and multiplication the set C[x] forms a ring. And then we can identify several important subrings by restricting the coefficients in various ways. For example, Z[x] is the subring of polynomials with coefficients in the ring Z of integers, Q[x] is the subring of polynomials with coefficients in the field Q of rational numbers, and ℜ[x] is the subring of polynomials with coefficients in the field ℜ of real numbers. Each of these collections of polynomials has interesting mathematical properties.
If P(x) is a nonzero polynomial in Z[x] it may happen that P(x) can be written as a product of polynomials in Z[x]. Some examples of this are
x2 - 2x - 15 = (x + 3)(x - 5), and x8 - 1 = (x - 1)(x + 1)(x2 + 1)(x4 + 1).
We say that a nonzero polynomial P(x) in Z[x] is irreducible if in any factorization of the polynomial as P(x) = Q(x)R(x) one of the factors is equal to 1 or -1. Irreducible polynomials in Z[x] are similar to prime numbers in the ring Z. This is because every polynomial in Z[x] can be factored in essentially one way into a product of irreducible polynomials. In general it is not easy to decide if a given polynomial in Z[x] is irreducible.
Each polynomial in C[x] determines a polynomial function from the field C to the field C. If P(x) is a polynomial in C[x] it determines the polynomial function that sends the complex number to the complex number P(). Usually a polynomial function is simply called a polynomial, and it is clear from the context if the polynomial is to be regarded as a function or as an element of a ring of polynomials. Because each polynomial P(x) determines a function in this way, we may ask about the set of complex numbers such that P() = 0. This set is called the set of roots or the set of zeros of the polynomial. One of the most important results about polynomials is the fundamental theorem of algebra. This theorem asserts that every polynomial in C[x] with positive degree has a root in C. The first rigorous proof of the fundamental theorem of algebra was given by Gauss. Notice that the same result does not hold if the field C is replaced by ℜ: the polynomial x2 + 1 has positive degree and coefficients in ℜ, but it does not have a root in ℜ. Also, if P(x) is a polynomial in C[x] with positive degree and 1 in C is a root, then P(x) factors in the ring C[x] as P(x) = (x - 1)Q(x). If Q(x) has positive degree we can apply the fundamental theorem of algebra again to Q. If 2 is a root of Q then we get the factorization P(x) = (x - 1)(x - 2)R(x). Continuing in this manner, we find that a polynomial of degree N in C[x] can be written as
P(x) = c0 + c1x + c2x2 + c3x3 + ... + cNxN
= cN(x - 1)(x - 2) ... (x - N),
where the complex numbers 1, 2, ... , N are the roots of P(x). Of course we cannot conclude that the roots are distinct. For example, we have the factorization
x6 + 2x5 + 2x4 - 2x2 - 2x - 1 = (x - 1)(x + 1) (x - )(x - )(x - ¯)(x - ¯),
where = -½ + i½3 and ¯ = -½ - i½3. In this case we say that and ¯ are roots of multiplicity two. More generally, if a complex number appears exactly M times among the roots of the polynomial P(x), then we say that P has a root of multiplicity M at . Thus a polynomial of degree N 1 in C[x] has exactly N roots in C provided the roots are counted with multiplicity.
Polynomials in the ring ℜ[x] are often used as polynomial functions to approximate more general continuous functions. Let [u, v] ⊆ ℜ be a closed interval in ℜ and let ƒ:[u, v] ℜ be a continuous function. Then the Weierstrass approximation theorem states that for every ε > 0 there exists a polynomial P(x) in ℜ[x] such that
|ƒ(x) - P(x)| < ε
for all real numbers x with u x v. Thus every continuous function defined on a finite, closed interval in ℜ can be uniformly approximated by a polynomial. In the special case [u,v] = [0,1] there is a particularly simple type of polynomial that can be used to uniformly approximate an arbitrary continuous function. Let ƒ: [0,1] ℜ be continuous and then define the Nth Bernstein polynomial associated to ƒ by
It can be shown that for every ε > 0 there exists an integer M such that
|ƒ(x) - BN(x)| < ε
for all real numbers x with 0 x 1 and all integers N M. Alternatively, the sequence BN(x) associated to ƒ converges uniformly to ƒ on [0, 1] as N .
This is the complete article, containing 1,093 words
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