If I have a general d-dimension cubic form $C\_0(x)$ with coefficients in $\mathbb{R}$ is it possible to find a cubic form $C\_1(x)$ with coefficients in $\mathbb{R}$ such that for all $x\in\mathbb{R}^{d}$ the following are satisfied:


i) $C\_1(x)=Q(x)L(x)$ where $Q(x)$ and $L(x)$ are quadratic and linear forms respectivly.
ii)$\left|C\_0(x)-C\_1(x)\right|<\delta$ for some $\delta>0$. 


Or is it possible that one can approximate $C\_0(x)$ by $C\_1(x)$ in some other way? 



Is it possible to approximate a general cubic form by one which factorises?
I would expect not. The reason being that if this is anything like true (products of quadratic forms and and linear forms being dense in the ordinary "strong" or norm topology - presumably what you mean) then more would be true than that. So for a proof one could either write down an equation that the coefficients of such products would satisfy (making them not Zariski dense). Or say something about the products being the image of an algebraic morphism, hence a constructible set. If this is even Zariski dense, it is huge. 


