<s>[INST] Determinant properties Prove without expanding:
\begin{equation}\begin{vmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3 & c^3\end{vmatrix} = (ab + ac + bc)(b - a)(c - a)(c - b)\end{equation}


*I tried to zero some elements and expand until I reach the Right hand side.

*Also tried C1-C3, C2-C3 then decompose the determinant into two determinants and taking common factors. But I couldn't get (ab + ac + bc) part.

*I can only use the properties shown here http://www.vitutor.com/alg/determinants/properties_determinants.html
 [/INST] \begin{gather*}
\begin{bmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3&c^3\end{bmatrix} \\
= \begin{bmatrix}1&0&0\\a^2&b^2-a^2&c^2-a^2\\a^3&b^3-a^3&c^3 - a^3\end{bmatrix} \\
= \begin{bmatrix}b^2-a^2&c^2-a^2\\b^3-a^3&c^3 - a^3\end{bmatrix} \\
= (b^2-a^2)(c^3 - a^3)-(b^3-a^3)(c^2 - a^2)
= (b-a)(c-a)[(b+a)(c^2+ac+a^2)-(c+a)(b^2+ab+a^2)] \\
= (ab+ac+bc)(b-a)(c-a)(c-b)\\
= R.H.S
\end{gather*}
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