Checking if 8 points in the projective plane lie on a singular cubic. I need to check if 8 points of $\mathbb{P}^2$ (over a finite field) lie on a singular cubic with one of them a double point.
I know that to check if a point is singular we suffice to compute the partial derivatives in it and look if they are all 0.
I also know I can use the Veronese embedding to prove that 10 points of $\mathbb{P}^2$ are independent with regard to cubics, i.e. don't lie in one, if and only if the matrix that has as rows the evaluation of monomials of degree 3 in a given point has rank 10.
However I'm not sure how to unite the two concepts to find an easy (as in &quot;easy to feed to a computer to check&quot;) condition to check if there is a singular cubic passing through the 8 points and having a singularity in one of them.
My idea is to use a matrix that has the following rows:

rows 1-7 evaluation of monomials of degree 3 in each of 7 points
rows 8-10 the partial derivatives to such monomials evaluated in the 8-th point

Repete 8 times changing the point we want to check is singular.
But I can't find a proper proof that this does actually work.
At the end the method I proposed actually works
At the end the method I proposed actually works