<s>[INST] Expressing a determinant in the form $k(a^3 + b^3 + c^3 - 3abc)^n$ I have been given a determinant $\begin{vmatrix} bc - a^2 & ac - b^2 & ab - c^2 \\ ac - b^2 & ab - c^2 & bc - a^2 \\ ab - c^2 & bc - a^2 & ac - b^2\end{vmatrix}$ and I want to compute it in the form $k(a^3 + b^3 + c^3 - 3abc)^n$
Now I know that for determinants like these this property exists that $\begin{vmatrix} a & b & c \\ b & c & a  \\ c & a & b\end{vmatrix}$ can be written as $-(a^3 + b^3 + c^3 - 3abc)$ but I am not able to figure out a way to do some operations so as to express the matrix above in this form. Ofcourse I could compute the entire determinant and then possibly factories it or so, but that process looks very tedious and definitely doesn't seem like the right approach to it. Any hints/explanations are appreciated on how to approach this.
 [/INST] The determinant is fully symmetric and homogeneous of degree $6$, and vanishes if $c=-a-b$ since then all entries are $-a^2-ab-b^2$, and also if $a=b=c$ since then all entries are $0$, so is divisible by both $a+b+c$ and$$\tfrac12((a-b)^2+(b-c)^2+(c-a)^2)=a^2+b^2+c^2-ab-bc-ca,$$and hence by their product, $a^3+b^3+c^3-3abc$. The determinant is also invariant under changing the signs of all variables, so it's of the desired form with $n=2$. Then $k$ is just the determinant for $a=1,\,b=c=0$.
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