Let $k$ be a field, $f(x,y)\in \bar{k}[x,y]$. I claim that
$f(x,y) = (x-a)g(x,y) +f(a,y)$.
I worked on it but could not show the equality. After that, I want to show that $$f(a,b) = 0 \implies f(x,y) = (x-a)g_1(x,y)+(y-b)g_2(x,y).$$
So I need some help to prove/disprove these. Any hints would be helpful.