Multivariate factor theorem
The Multivariable Factor Theorem states that If is a polynomial and there is a polynomial such that for [b]all[\b] 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!