Multivariate factor theorem
The Multivariable Factor Theorem states that If is a polynomial and there is a polynomial
such that
for all
then we can write
for some polynomial
Proof:
Assume that for all
. We'll treat
as a constant, so that
is constant with respect to
If we divide by
using polynomial long division, so that we have
Since we're treating as a constant,
is a monic, linear polynomial in
So, either
is the zero polynomial, in which case it has no terms with
or it has lower degree in
than
This means that
will itself be a polynomial in
Now, if we set in our equation, it becomes
It follows that
So for any
and so
is the zero polynomial!