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Remainder factor theorem states that if the polynomial $f(x)$ is divided by $(x-c)$ then the remainder is $f(c)$, how does this work for multivariable polynomials?

I am confused by this: enter image description here

How does showing $f(x,x)=0$ prove $(x-y)$ divides $f(x,y)$ for all $x,y$?

Tian Vlašić
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bobbyJames
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    See for example my answer here, yours is the case $n=2,x_1=x,x_2=y,f(x_1)=x_2$. Basically, you show that $(y-x) \mid f(x,y) - f(x,x)$ for any bivariate polynomial $f(x,y)$, then if $f(x,x) = 0 \ldots$ – dxiv Aug 02 '23 at 02:27

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