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

Options

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

Answer

To solve the given problem step by step, we need to analyze the function \( f(x) \) and its determinant.

### Step 1: Understand the Given Information
We are given:
- \( \sum a^2 = -2 \), which implies \( a^2 + b^2 + c^2 = -2 \).
- The function \( f(x) \) is defined as:
\[ f(x) = \begin{vmatrix} (1 + a^2)x & (1 + b^2)x & (1 + c^2)x \\ (1 + a^2)x & 1 + b^2x & (1 + c^2)x \\ (1 + a^2)x & (1 + b^2)x & 1 + c^2x \end{vmatrix} \]

### Step 2: Simplify the Determinant
We can simplify the determinant by performing row operations. We will subtract the first row from the second and third rows.

\[ f(x) = \begin{vmatrix} (1 + a^2)x & (1 + b^2)x & (1 + c^2)x \\ 0 & (1 + b^2)x - (1 + a^2)x & (1 + c^2)x - (1 + a^2)x \\ 0 & (1 + b^2)x - (1 + a^2)x & (1 + c^2)x - (1 + a^2)x \end{vmatrix} \]

This results in:
\[ f(x) = \begin{vmatrix} (1 + a^2)x & (1 + b^2)x & (1 + c^2)x \\ 0 & (b^2 - a^2)x & (c^2 - a^2)x \\ 0 & (b^2 - a^2)x & (c^2 - a^2)x \end{vmatrix} \]

### Step 3: Evaluate the Determinant
Now, we can see that the second and third rows are identical. Therefore, the determinant is zero:
\[ f(x) = 0 \]

### Step 4: Determine the Degree of the Polynomial
Since \( f(x) = 0 \), it is a polynomial of degree \( 0 \).

### Final Answer
The degree of the polynomial \( f(x) \) is \( 0 \).