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!