Factor Theorem
Revision as of 14:38, 15 November 2007 by 10000th User (talk | contribs) (Found a proof I provided a long time ago, a gem hidden in the forums)
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If is a factor of
, then
, where
is a polynomial with
. Then
.
Now suppose that .
Apply division algorithm to get , where
is a polynomial with
and
is the remainder polynomial such that
.
This means that can be at most a constant polynomial.
Substitute and get
.
But is a constant polynomial and so
for all
.
Therefore, , which shows that
is a factor of
.