3. Polynomials

A polynomial or rational function is an expression of the form

with given real or complex numbers c_i 0, where the integer n is called the degree of f(x). Occasionally, it will be expedient to refer to real or complex numbers as polynomials of degree zero or constant polynomials. The root or zero of the polynomial f(x) is the number a for which f(a) = 0. The fundamental statement about the existence of roots is the Fundamental Theorem of Algebra*:

Within the range of all real and complex numbers, every polynomial f(x) of degree n 0 has n roots. If a₁···a_n are the roots of f(x), f(x) has the decomposition into linear factors

(3.1)

Not all of the n roots need be different. If k of these roots are a, you call a a k-fold or multiple root of f(x). The decomposition then includes the power (x-a)^k of the linear factor (x-a).

*The proof of this fundamental theorem is beyond the framework of this book.

Theorem 1

If f(x) and g(x) are two polynomials without common root, there exist two further polynomials j(x) and y(x) for which

f(x)j(x) + g(x)y(x) = 1.

Proof: One may assume without affecting generality that

degree f(x) degree g(x).

If g(x) were to be of degree 0, that is, reduced to a number c, the assumption of Theorem 1 tells that c 0. Then Theorem 1 is certainly true; you just set j(x)=0, y(x)= 1/c.

We now execute the proof by induction and assume accordingly that Theorem 1 holds for all pairs f₁(x), g₁(x) of polynomials without a common root for which

degree f₁(x) degree g₁(x), degree g₁(x) < degree g(x),

whence the induction parameter is here the lowest degree of a pair of polynomials.

If you divide f(x) by g(x), you obtain

f(x) = q(x)g(x) + r(x) with degree r(x) < degree g(x). (3.2)

The two polynomials g(x) and r(x) do not have a common root, for if you had g(a) = r(a) = 0, Equation (3.2) would yield f(a) = 0, that is, in contradiction of the assumption referring to common roots, f(x) and g(x) would have the common root x = a , whence you can now apply the induction argument to the polynomials g(x) and r(x).

There exist now two polynomials j₁(x) and y₁(x) such that

g(x)j₁(x) + r(x)y₁(x) = 1.

Using (3.2), replace r(x) by f(x) - q(x)g(x) and find

g(x)j₁(x) + [f(x) - q(x)g(x)]y₁(x) = 1

or

f(x)y₁(x) + g(x)[j₁(x) - q(x)y₁(x)] = 1,

whence the assumption has been proved for f(x) and g(x).

A polynomial in the variables x, y, ···, z has the form

where you must sum over a certain finite number of systems r, s, ···, t of non-negative integers. The coefficients c_{r, s, ··· , t} are given real or complex numbers. If c_{r, s, ··· , t} 0, you call the sum of exponents r + s + ··· + t of the term c_r,s,···,tx^ry^s ···, z^t the degree of this term. The degree of a polynomial is the highest degree occurring in a polynomial. A homogeneous polynomial or form has only terms of the same degree. A polynomial is homogeneous of degree k with respect to one variable if this variable occurs in all terms at that degree.

Two polynomials in the same variables are identical if the coefficients of the same products of the powers of the variable are the same. Naturally, identical polynomials have the same value whatever the values of the variables. The inverse of this statement is the

Polynomial Identity Theorem

If the polynomials f(x,y,···, z) and g(x,y,···, z) have always the same value whatever values have the variables, they are identical.

Proof: Form the difference

d(x,y,···, z) = f(x,y,···, z) - g(x,y,···, z),

which vanishes by assumption. It is to be shown that d(x,y,···, z) is the null-polynomial, that is, that all coefficients vanish. In order to confirm this, apply induction with respect to the number of variables. In the case n = 1, the statement is correct, because the polynomial d(x), not all coefficients of which vanish, has only a finite number of zeros and can therefore not vanish for all its values. Assume now that the statement has been proved already for polynomials with at most n - 1 variables, whence it cannot vanish for each value of the variable. Ordering d(x,y,···, z) according to the powers of x yields

d(x,y,···, z) = x^ma_m(y,···, z) + x^m-1a_m-1(y,···, z) + ··· + a₀(y,···, z)

involving the polynomials a_i(y,···, z) with at most n - 1 variables. Since d(x,y,···,z) vanishes for arbitrarily fixed values of y,···, z for all values of x, all the a_i(y,···, z) must vanish for any values of y,···, z.Hence, by the induction assumption, all coefficients in the a_i(y,···, z) vanish, whence d(x,y,···, z) is the null-polynomial.

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