Calculate the zeros of a polynomial in finite fields

Question

How to calculate polynomial zeros ${y^3} - \left( {{x^2}} \right)y + x\left( {{y^2}} \right)$ in ${\mathbb{F}_5}\left[ {x,y} \right]$?

Answer 1

The polynomial is homogeneous of degree $3$. Each non-zero solution lies on a line of solutions.

Thus it is sufficient to consider $y=1$, the partially evaluated polynomial $1+x-x^2$ has the solution $(x,y)=(3,1)$, thus also $(1,2)$, so that $y-2x$ is a linear factor of the given polynomial. Extracting this factor gives the full factorization as $$ y^3+y^2x-yx^2=y(y^2-2y(2x)+(2x)^2)=y(y-2x)^2\pmod5 $$