Multivariate factor theorem
[b] The Multivariable Factor Theorem [b] states that If is a polynomial and there is a polynomial
such that
for [b]all[/b]
then we can write
for some polynomial
[b] Proof:[/b]
Assume that for all
. We'll treat
[i]as a constant,[/i] 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!