If `a^(2)+b^(2)+c^(2)`=-2 and f(x)= `|{:(1+a^(2)x,(1+b^(2))x,(1+c^(2))x),((1+a^(2))x,1+b^(2)x,(1+c^(2))x),((1+a^(2))x,(1+b^(2))x,1+c^(2)x):}|` then f(x) is a polynomial of degree

Options

A. 3
B. 2
C. 1
D. 0

Answer

To solve the problem, we need to find the degree of the polynomial \( f(x) \) defined by the determinant:

\[ f(x) = \left| \begin{array}{ccc} 1 + a^2 x & 1 + b^2 x & 1 + c^2 x \\ 1 + a^2 x & 1 + b^2 x & 1 + c^2 x \\ 1 + a^2 x & 1 + b^2 x & 1 + c^2 x \end{array} \right| \]

Given that \( a^2 + b^2 + c^2 = -2 \), we will simplify the determinant step by step.

### Step 1: Write the Determinant

We start by writing the determinant explicitly:

\[ f(x) = \left| \begin{array}{ccc} 1 + a^2 x & 1 + b^2 x & 1 + c^2 x \\ 1 + a^2 x & 1 + b^2 x & 1 + c^2 x \\ 1 + a^2 x & 1 + b^2 x & 1 + c^2 x \end{array} \right| \]

### Step 2: Simplify the Determinant

Notice that all rows of the determinant are identical. Therefore, the determinant of a matrix with identical rows is zero:

\[ f(x) = 0 \]

### Step 3: Analyze the Result

Since \( f(x) = 0 \), it implies that \( f(x) \) is a constant function (specifically, the constant zero). The degree of a constant function is defined as 0.

### Conclusion

Thus, the degree of the polynomial \( f(x) \) is:

\[ \text{Degree of } f(x) = 0 \]

### Final Answer

The degree of \( f(x) \) is **0**.

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