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$x^{2i}+y^{2j}$ isn't usually divisible by $x^2+y^2$.

$x^2(y^2+y+1)+y^2(x^2+x+2)$ is an example showing that odd powers can be used as well.

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$x^{2i}+y^{2j}$ isn't usually divisible by $x^2+y^2$.