260 Chapter 5- Show that w = Az + Bz + C, with IAI i= IBI transforms circles into
ellipses. - By examining the conditions D 1 = 0, ID 2 I i= ID 3 I determine all possible
types of general bilinear transformations that are one-to-one in G =
{z: Lz+Mz+N#O}.
5.14 THE POLYNOMIAL FUNCTION w = P(z) =
(l~o + aiz + ... + anzn
A function defined by a polynomial in a complex variable z, with constantcomplex coefficients a 0 , ai, ... , an, is called a polynomial function (in
one variable). It will be assumed that an i= 0, and then the nonnegative
integer n is called the degree of the polynomial. If ri = 0, P(z) reduces to
a constant (a constant polynomial), and if n = 1, we have w = ao + aiz,
which is the linear function discussed in Section 5.3.
According to the fundamental theorem of algebra, to be proved later,
for n ~ l the polynomial P(z) has at least one root. If P(o: 1 ) = 0, then as
a consequence of the "remainder theorem" of elementary algebra, we haveP(z) = (z -0:1)P1(z) (5.14-1)where P 1 (z) is a polynomial of degree n - 1. If n ~ 2 and P 1 (o: 2 ) = O,
we get, similarly,
(5.14-2)with P2(z) a polynomial of degree n-2. Substitution of (5.14-2) in (5.14-1)
gives
By a continuation of the same argument we arrive at the factorization
of the polynomial in the form(5.14-3)which is unique, except for the order of the factors, and where the zeros or
roots 0:1, 0:2, ••• , O:n are not necessarily distinct. If 0:1 appears hi times, 0:2
appears h 2 times, ... , and O:p appears hp times, where hi +h2+· ··+hp= n,
then (5.14-3) can be written as
P(z) = an(z -o:1)h1(z - o:2)h2 ... (z - o:p)hp
and we say that o: 1 is a zero or root of order h 1 (or of multiplicity h 1 ), that
a2 is a zero or root of order h2, and so on.