Question

The total number of distinct $x\in R$ for which $\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10$ is

Answer 1

$\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10$
$x^3\left|\begin{array}{ccc}1& 1& 1\\ 2& 4& 1\\ 3& 9& 1\end{array}\right|+x^6\left|\begin{array}{ccc}1& 1& 1\\ 2& 4& 8\\ 3& 9& 27\end{array}\right|=10$
$R_3\to R_3-R_2$
$C_3\to C_3-C_2$
$R_2\to R_2-R_1$
$C_2\to C_2-C_1$
$x^3\left|\begin{array}{ccc}1& 1& 1\\ 1& 3& 0\\ 1& 5& 0\end{array}\right|+x^6\left|\begin{array}{ccc}1& 0& 0\\ 2& 2& 4\\ 3& 6& 18\end{array}\right|$
$2x^6+2x^3-10=0$
$6x^6+x^3-5=0$
$(x^3+1)(6x^3-5)=0$