If x, y, z are different and `Delta=|[x,x^2, 1+x^3],[y, y^2,1+y^3],[z, z^2, 1+z^3]|=0`, then show that `1+xyz=0`

Explanation

To solve the problem, we need to show that if the determinant \(\Delta = \begin{vmatrix} x & x^2 & 1 + x^3 \\ y & y^2 & 1 + y^3 \\ z & z^2 & 1 + z^3 \end{vmatrix} = 0\), then it follows that \(1 + xyz = 0\).

### Step-by-Step Solution:

1. **Write the Determinant**:
\[ \Delta = \begin{vmatrix} x & x^2 & 1 + x^3 \\ y & y^2 & 1 + y^3 \\ z & z^2 & 1 + z^3 \end{vmatrix} \]
...