Determine all functions $k$ such that $k(x,y,z)≠0$ if $x≠0$ Let u consider the function:
$$k(x,y,z)=c₀+c₁x+c₂y+c₃z+c₄x²+c₅y²+c₆z²+c₇xy+c₈xz+c₉yz$$
My question is: 
Determine all functions $k$ such that $k(x,y,z)≠0$ if $x≠0$.
Here $(c_{i})_{0≤i≤9}∈ℝ$ are real constants.
 A: We distinguish between $c_0=0$ and $c_0\ne 0$.
Assume $c_0=0$. Then $k(x,0,0)=c_1x+c_4x^2=x(c_1+c_4x)$. To avoid a nonzero root, we must have
$$\tag1 c_4=0, c_1\ne0\qquad\text{or}\qquad c_1=0,c_4\ne0. $$
Assume $c_0\ne 0$. Then $k(x,0,0)=c_0+c_1x+c_4x^2$. If $c_4=0$, we need $c_1=0$ to avoid a root $-\frac{c_0}{c_1}$. If $c_4\ne 0$, we need a negative discriminant to avoid a real root (which cannot be at $x=0$ anyway), i.e. we must have
$$\tag 2c_1=c_4=0\qquad\text{or}\qquad c_1^2<4c_0c_4. $$
Next note that
$$ k(x,y,0)=(c_0+c_2y+c_5y^2)+(c_1+c_7y) x + c_4x^2.$$
For any fixed $y$ we need the conditions corresponding to (1) and (2) to hold that is: whenever $c_0+c_2y+c_5y^2=0$ we must have $c_4=0, c_1+c_7y\ne0$ or $c_4\ne0, c_1+c_7y=0$. And whenever $c_0+c_2y+c_5y^2\ne0$ we must have $c_1+c_7y=c_4=0$ or $(c_1+c_7y)^2<4(c_0+c_1y+c_5y^2)c_4$. You can find many conditions from this; for example "most" combinations of $c_2,c_5$ will lead to $c_0+c_2y+c_5y^2\ne 0$ for almost all $y$ and $=0$ for zero, one or two values of $y$. 
After that, you can investigate $k(x,y,ty)$ for fixed $t$ in a similar fashion
