Let $\nu$ be the map $\mathbb{P}^1 \to \mathbb{P}^2$ defined by $\nu([X_0,X_1])=[A_0(X_0,X_1),A_1(X_0,X_1),A_2(X_0,X_1)]$ where the $(A_i)$ are homogenous polynomials of degree 3, without common zeros.
To prove the image of this map is included in a cubic curve of $\mathbb{P}^2$, we can look at the pullback of the map $\nu$ that gives a linear map from the space of homogenous polynomials of degree 3 on $\mathbb{P}^2$ to the space of homogenous polynomials of degree 9 on $\mathbb{P}^1$ (this comes from a hint in an exercise of Harris' book Algebraic Geometry). It is then sufficient to prove that this pullback is not surjective since the two spaces have dimension 10.
I have some difficulty to prove that by a simple (meaning not using powerful theorems of elimination theory) method. Any hint about this hint ? (Harris seems to imply in his hint that it is quite obvious : perhaps there is something very simple to see that I am missing)
PS: I posted below a simple answer I just found. If someone can check it, I will be grateful !