let $ f(x,y,z) \in \mathbb{R}[x, y, z] $ such that :
$$f(x,y,z)=x^{2}+y^{2}+z^{2}-y x-z x-zy$$
Is $Z(f(x,y,z))$ affine variety in $\mathbb{A^3}_{\mathbb{R}}$ ?
i think we can show $Z(f)$ is irreducible .If we can show $I=\left(x^{2}+y^{2}+z^{2}-y x-z x-z y\right)$ is the kernel of the some ring homomorphism $\mathbb{R}[x, y, z] \rightarrow \mathbb{R}[x]$ Since $\mathbb{R}[x]$ is a domain, then ideal $I$ is prime.(note that $Z(f)=\left\{P \in \mathbb{A^3}_{\mathbb{R}} \mid f(P)=0\right. \}$)