My first attempt : Tried to solve by polynomial formulas but can't proceed after few steps. Second attempt : Tried by vectors but found nothing useful.
2026-03-25 18:50:13.1774464613
For which values of $k$ can $( x +y +z)^2 + k(x^2 +y^2 +z^2)$ be resolved into linear rational factors?
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If $ax^2+by^2+cz^2+2ryz+2sxz+2txy$ is a homogeneous quadratic in three variables, then it can only split into linear factors if the determinant $$\begin{vmatrix}a&r&s\\r&b&t\\s&t&c\end{vmatrix}=0.$$ Here, that determinant is $$\begin{vmatrix}k+1&1&1\\1&k+1&1\\1&1&k+1\end{vmatrix}.$$