So when they say 'ci' they aren't using i to mean sqrt(-1)? The use of 'i' has really confused me.
No; all the subscripts are just that: counters which have been subscripted.
EDIT: A question I forgot to mention in my OP, how is there a zero, c, that is not c1,c2....cn?
The proof says exactly that: any zero "c" must
be one of the "n" zeroes denoted "ci
" for i = 1 through i = n.
I still can't see how this proof shows us how every polynomial has exactly as many factors as the degree of the polynomial?
The proof starts with a zero, "c", and then shows that this zero must
be equal to one of the n zeroes, ci
. Since any zero c must be one of the n zeroes denoted in the factored form, then the polynomial can't have any more than n zeroes. Since the polynomial has n factors and thus n corresponding zeroes, then the polynomial has at least n zeroes. Since the polynomial has "at least" n zeroes but "no more than" n zeroes, then the polynomial must have exactly n zeroes.