Question

Factors of $\Delta =\left|\begin{array}{ccc}{x}^3& y^3& z^3\\ yz& zx& xy\\ 1& 1& 1\end{array}\right|$ are

• A. $y-z$
• B. $z-x$
• C. $x-y$
• D. $x^2+y^2+z^2+yz+zx+xy$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $y-z$
B $z-x$
C $x-y$
D $x^2+y^2+z^2+yz+zx+xy$

$\Delta =\left|\begin{array}{ccc}{x}^3& y^3& z^3\\ yz& zx& xy\\ 1& 1& 1\end{array}\right|$
Applying $C_2\to C_2-C_1,C_3\to C_3-C_1$
$\Delta =\left|\begin{array}{ccc}{x}^3& y^3-x^3& z^3-x^3\\ yz& z(x-y)& x(y-z)\\ 1& 0& 0\end{array}\right|=(y-x)(z-y){\Delta }_1$
Where,
${\Delta }_1=\left|\begin{array}{cc}{y}^2+x^2+yx& z^2+y^2+zy\\ -z& -x\end{array}\right|=(z-x)(x^2+y^2+z^2+yz+zx+xy)$
Therefore,
$\Delta =(y-x)(z-y)(z-x)(x^2+y^2+z^2+yz+zx+xy)$