Page 84 - Calculus with Complex Numbers

Fu nd am ental theo rem of

a I ge b ra

8 . I Z e ros
We call the point c a zero of the function flz) if .J(c) = 0. For example the zeros
of sin z are at z = n:r for n = 0 +1 +2 . . . .
We deline the order or multiplicity of a zero as follows. Suppose flz) has Taylor
expansion

at z = c. We say c is a zero of order n if ao = tz1 = . . . = tzn-l = 0, but an # 0.
Hquivalently, if /'(c) = ./''(c) = . . . = ./*(n-1) (c) = 0, but /'(n) (c) # 0. A zero of
.
.
order 1 is called a simple zero a zero of order 2 is called a double zero etc. For
example, the zeros of f (z) = sin z are all simple since f (z) = cos z = ulu 1 at
z = nn'. However, for example, g (z) = z sin z has a double zero at z = 0 since
the M aclaurin expansion is

Theorem 1 (Fundamental theorem of algebra) Hvel'y polynomial of degree
n with complex coeflicients has n zeros in the complex plane taking account of
multiplicity.

Case n = 2 Hvel'y quadratic polynomial p (z) with complex coeflicients has
2 roots, possibly coincident. The case of coincident roots is when p (z) is a perfect
square taking the form

plz) = Atz - .P)2

therefore p (z) has a double zero at z = B .