TITLE: Factoring polynomial in two variables.
QUESTION [7 upvotes]: Given a polynomial $P(x,y)$ I would like to know what the criteria are for factoring out linear factors.
For instance, in one variable, if $Q(a) = 0$, then one may say $Q(x) = (x-a)R(x)$. In two variables this is not true, as shown by $P(x,y) = x^2+y^2$ one has $P(0,0)=0$ but one cannot factor out anything.
When can one factor out a linear factor from a polynomial in two variables?

REPLY [3 votes]: The Factor Theorem remains applicable, for example
$\rm\qquad x-a\ |\ f(x) \iff f(a) = 0\ \ $ for $\rm\:a = by+c \in R[y]\:$ is
$\rm\qquad x-by-c\ |\ f(x,y)\iff f(by+c,\:y) = 0$
e.g. $\rm\ x-y\ |\ f(x) - f(y)\ \:$ since $\rm\ f(y)-f(y) = 0 $