<s>[INST] Factorise a matrix using the factor theorem Can someone check this please?
 $$
\begin{vmatrix}
x&y&z\\
x^2&y^2&z^2\\
x^3&y^3&z^3\\
\end{vmatrix}$$
$$C_2=C_2-C_1\implies\quad
\begin{vmatrix}
x&y-x&z\\
x^2&y^2-x^2&z^2\\
x^3&y^3-x^3&z^3\\
\end{vmatrix}$$
$$(y-x)
\begin{vmatrix}
x&1&z\\
x^2&y+x&z^2\\
x^3&y^2+xy+x^2&z^3\\
\end{vmatrix}$$
$$(y-x)(z-x)
\begin{vmatrix}
x&1&1\\
x^2&y+x&z+x\\
x^3&y^2+xy+x^2&z^2+xz+x^2\\
\end{vmatrix}$$
$$R_2=R_2-xR_1\implies\quad
(y-x)(z-x)
\begin{vmatrix}
x&1&1\\
0&y&z\\
x^3&y^2+xy+x^2&z^2+xz+x^2\\
\end{vmatrix}$$
$$R_3=R_3-x^2R_1\implies\quad
(y-x)(z-x)
\begin{vmatrix}
x&1&1\\
0&y&z\\
0&y^2+xy&z^2+xz\\
\end{vmatrix}$$
factor $x$$$\implies\quad
x(y-x)(z-x)
\begin{vmatrix}
1&1&1\\
0&y&z\\
0&y^2+xy&z^2+xz\\
\end{vmatrix}$$
$$\implies\quad x(y-x)(z-x)(yz^2-zy^2)$$
$$\implies\quad xyz(y-x)(z-x)(z-y)$$
Also I'd like practical tips on using the factor theorem for these types of questions. My understanding is that the determinant is $f(x,y,z)$ so if we hold $y$ and $z$ constant we could apply it somehow to $f(x)$ alone. I'm not that great spotting difference of squares etc and want a more fail safe alternative. Thanks in advance.
 [/INST] you are not factoring the matrix. you are using the properties of the determinants to simplify. for example, you could write your first step as 
$$AE = \pmatrix{x & y & z \\ x^2 & y^2 & z^2 \\ x^3 & y^3 & z^3} 
\pmatrix{1 & -1 & 0\\0&1&0\\0&0&1} =
\pmatrix{x & y-x & z \\ x^2 & y^2-x^2 & z^2 \\ x^3 & y^3 -x^3& z^3} = B
 $$
the $E$ matrices are called elementary column matrices and their determinants usually is the product of the diagonals. now use the product rule of the determinants $det(AE) = det(A) det(E)$ to conclude that $det(A) = det(B)$ now you start with $B$ and do further reductions.
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