Suppose n,m are natural numbers and
`f(x)=|(1,(1+x)^(m),(1+mx)^(mn)),((1+mx)^(n),1,(1+nx)^(mn)),((1+nx)^(m),(1+x)^(n),1)|`
constant term of the polynomial `f(x)` is

Options

A. 1
B. `m+n`
C. `m-n`
D. `0`

Answer

To find the constant term of the polynomial \( f(x) \) given by the determinant:

\[ f(x) = \begin{vmatrix} 1 & (1+x)^m & (1+mx)^{mn} \\ (1+mx)^{n} & 1 & (1+nx)^{mn} \\ (1+nx)^{m} & (1+x)^{n} & 1 \end{vmatrix} \]

we will evaluate the determinant and focus on the constant term.

### Step-by-Step Solution:

1. **Understanding the Determinant**:
The determinant is a function of \( x \) and is formed by three rows and three columns. Each entry in the determinant is a polynomial in \( x \).

2. **Identifying Constant Terms**:
The constant term in a polynomial is the term that does not involve \( x \). We need to find the contributions to the determinant that yield a constant term when expanded.

3. **Expanding the Determinant**:
We will expand the determinant using the properties of determinants. The determinant can be expanded as follows:

\[ f(x) = 1 \cdot \begin{vmatrix} 1 & (1+mx)^{mn} \\ (1+nx)^{m} & (1+x)^{n} \end{vmatrix} - (1+x)^m \cdot \begin{vmatrix} (1+mx)^{n} & (1+nx)^{mn} \\ (1+nx)^{m} & 1 \end{vmatrix} + (1+mx)^{mn} \cdot \begin{vmatrix} (1+mx)^{n} & 1 \\ (1+nx)^{m} & (1+x)^{n} \end{vmatrix} \]

4. **Finding the Constant Term**:
- The constant term from the first determinant is \( 1 \cdot 1 = 1 \).
- The second determinant will yield a constant term of \( 0 \) because it includes terms with \( x \) in both rows.
- The third determinant will also yield a constant term of \( 0 \) for similar reasons.

Therefore, the total constant term from the expansion is:

\[ 1 - 0 + 0 = 1 \]

5. **Conclusion**:
The constant term of the polynomial \( f(x) \) is \( 1 \).

### Final Answer:
The constant term of the polynomial \( f(x) \) is \( 1 \).