Since
is an integral domain we can use the division algorithm.
Now lets divide
This gives
but since
Is degree 1 r(x) must be degree zero or a constant this gives
Now just evaluate at alpha
So
Let be a field, let be a polynomial, and let .
Then I understand that if , then , this is obvious.
But it doesn't seem obvious to me that if , then . How can I show this? Is there some result about factorisation that implies this?
Any help would be appreciated.
for any polynomial we can write:
f(x) - f(a) = (x - a)q(x) (see the previous post).
if f(x) = (x - a)k(x), we get f(a) = (x - a)(k(x) - q(x)).
the LHS is a constant polynomial, and the RHS is a polynomial of degree ≥ 1, which is a contradiction....unless k(x) = q(x),
in which case we have f(a) = 0.