Factor Theorem
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Introduction
Theorem and Proof
Theorem: If is a polynomial, then
is a factor
iff
.
- Proof: 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
.