Page 89 - Calculus with Complex Numbers

0 I 2

Ffgure 8.5

8.4 Proof of the fundam ental theorem of algebra

Suppose that
(z) = anzn + Jn-lzn -1 + .- + J1z + Jo,
J'
where an # 0, is a polynomial of degree rl. Let

flz) = anzn,

and let yR be the contotlr Izl= R. Then on yR we have

.'? (c) tzn-lcn-l + . . . + tzlc + ao
f (z) anzn
IJn-lz''-l I + . . . + Itzlzl+ Itzol

S I a
nzn I

Itzn-l IA''-1 + . . . + Itzl I R + Itzol
IlnlAn

as R --> co. Therefore, we can choose R such that Iflz) I> Ig (z) Ifor all z eE yR .

lt follows by Rouché's theorem that p (z) = flz) + g (z) and f (z) have the
same number of zeros inside yR for this R. But flz) = anzn has n zeros inside
yR, all at z = 0. Hence also p (z) has n zeros inside yR, as required.

Exam ples