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):}|` the f(x) is a polynomial of degree

Options

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

Answer

To solve the problem, we need to determine 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 proceed step by step.

### Step 1: Rewrite the Determinant
The determinant can be rewritten as:

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

### Step 2: Apply Row Operations
We can simplify the determinant by performing row operations. Specifically, we can subtract the second row from the first and the third row from the second:

\[ f(x) = \left| \begin{array}{ccc} 0 & 0 & 0 \\ 1 + a^2 x & 1 + b^2 x & 1 + c^2 x \\ 1 + a^2 x & 1 + b^2 x & 1 + c^2 x \end{array} \right| \]

### Step 3: Evaluate the Determinant
Since we have two identical rows, the determinant evaluates to zero:

\[ f(x) = 0 \]

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

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

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