Let $k$ be an algebraically closed field of characteristic 0 and let $C\subset \mathbb{P}^2$ be the curve given by $f(x,y,z)=x^2y^2+x^2z^2+y^2z^2=0$. I have determined that $[0,0,1]\in \mathbb{P}^2$ is a singular point of $C$.
I would like to find the multiplicity of this singular point. I know the definition of multiplicity of plane curves in the affine plane (from Hartshorne p. 36). For example, for the plane curve $V(f(x,y,1))\subset\mathbb{A}^2$, the origin is a singular point of multiplicity 2.
How can I find the multiplicity of this curve in projective space? Do I need to rewrite $f$ as $f=f_0+f_1+f_2+\dots$ where $f_i$ is homogeneous of degree $i$? I think the multiplicity is either 4 or 2. Is this correct?