How to apply the remainder theorem for multi-variable polynomials?

Question

In our math class, we were taught that for a polynomial $f(x)$

$$f(\alpha) \equiv f(x) \pmod {x-\alpha} $$

That's all very well. But, what about polynomials in more than variable?

Specifically, how can I apply the remainder theorem for a polynomial:

$$f(x_1, x_2, x_3, \dots, x_n)$$

in $n$ variables?

Answer 1

Well, if it helps, you still have $f(x_1, x_2, \dots, \alpha, \dots, x_n) \equiv f(x_1, x_2, \dots, x_i, \dots, x_n) \pmod {x_i - \alpha}$, as you can consider $f$ as a polynomial in $x_i$.