Value of `|{:(1+x_(1),,1+x_(1)x,,1+x_(1)x^(2)),(1+x_(2),,1+x_(2)x,,1+x_(2)x^(2)),(1+x_(3),,1+x_(3)x,,1+x_(3)x^(2)):}|` depends upon

Options

A. x only
B. `x_(1)`only
C. `x_(2)`only
D. none of these

Explanation

To find the value of the determinant

\[ D = \begin{vmatrix} 1 + x_1 & 1 + x_1 x & 1 + x_1 x^2 \\ 1 + x_2 & 1 + x_2 x & 1 + x_2 x^2 \\ 1 + x_3 & 1 + x_3 x & 1 + x_3 x^2 \end{vmatrix} \]

we will perform some operations on the columns to simplify it.

### Step 1: Column Operations
We will perform the following column operations:
- Replace \( C_1 \) with \( C_1 - C_2 \)
- Replace \( C_2 \) with \( C_2 - C_3 \)

This gives us:

\[ D = \begin{vmatrix} (1 + x_1) - (1 + x_1 x) & (1 + x_1 x) - (1 + x_1 x^2) & 1 + x_1 x^2 \\ (1 + x_2) - (1 + x_2 x) & (1 + x_2 x) - (1 + x_2 x^2) & 1 + x_2 x^2 \\ (1 + x_3) - (1 + x_3 x) & (1 + x_3 x) - (1 + x_3 x^2) & 1 + x_3 x^2 \end{vmatrix} \]

### Step 2: Simplifying the Determinant
Now simplifying the first two columns:

\[ D = \begin{vmatrix} x_1(1 - x) & x_1 x(1 - x) & 1 + x_1 x^2 \\ x_2(1 - x) & x_2 x(1 - x) & 1 + x_2 x^2 \\ x_3(1 - x) & x_3 x(1 - x) & 1 + x_3 x^2 \end{vmatrix} \]

### Step 3: Factor Out Common Terms
We can factor out \( (1 - x) \) from the first two columns:

\[ D = (1 - x)^2 \begin{vmatrix} x_1 & x_1 x & 1 + x_1 x^2 \\ x_2 & x_2 x & 1 + x_2 x^2 \\ x_3 & x_3 x & 1 + x_3 x^2 \end{vmatrix} \]

### Step 4: Further Simplification
Next, we can notice that the first two columns are proportional to \( x_1, x_2, x_3 \) and \( x_1 x, x_2 x, x_3 x \). So we can factor out \( x_1, x_2, x_3 \) from the first two columns:

\[ D = (1 - x)^2 x_1 x_2 x_3 \begin{vmatrix} 1 & x & 1 + x^2 \\ 1 & x & 1 + x^2 \\ 1 & x & 1 + x^2 \end{vmatrix} \]

### Step 5: Identical Rows
Now, we can see that the last determinant has identical rows, which means it evaluates to zero:

\[ D = 0 \]

### Conclusion
Thus, the value of the determinant depends on the values of \( x_1, x_2, x_3 \) and \( x \). Since the determinant evaluates to zero, we can conclude that the value of the determinant does not depend on any specific variable.