Key Concept: Properties of Determinants and Polynomial Degree

This problem requires us to evaluate a determinant, $f(x)$, and then determine its degree as a polynomial in $x$. The key to solving such problems efficiently lies in using the properties of determinants, specifically row and column operations, to simplify the expression before expansion. These operations do not change the value of the determinant but can transform it into a form that is much easier to compute. The degree of a polynomial is the highest power of the variable (in this case, $x$) present in its simplified form.

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Step 1: Simplify the Determinant using Column Operations

Our goal is to simplify the entries of the determinant, especially by leveraging the given condition $a^2 + b^2 + c^2 = -2$.

The given function is: $f\left( x \right) = \left| {\begin{matrix} {1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \\ {\left( {1 + {a^2}} \right)x} & {1 + {b^2}x} & {\left( {1 + {c^2}} \right)x} \\ {\left( {1 + {a^2}} \right)x} & {\left( {1 + {b^2}} \right)x} & {1 + {c^2}x} \\ \end{matrix} } \right|$

Operation: Apply the column operation $C_1 \to C_1 + C_2 + C_3$. Explanation: This operation replaces the first column ($C_1$) with the sum of all three columns ($C_1 + C_2 + C_3$). This is a strategic move often employed when elements across columns share similar structures or when we want to introduce a common factor. Observe that each term in the first column will now include $1 + (a^2+b^2+c^2+2)x$ if we sum them up.

Let's compute the new entries for the first column: For the first row, $C_{11} \to (1 + a^2x) + (1 + b^2)x + (1 + c^2)x = 1 + a^2x + x + b^2x + x + c^2x = 1 + (a^2+b^2+c^2+2)x$. For the second row, $C_{21} \to (1 + a^2)x + (1 + b^2x) + (1 + c^2)x = x + a^2x + 1 + b^2x + x + c^2x = 1 + (a^2+b^2+c^2+2)x$. For the third row, $C_{31} \to (1 + a^2)x + (1 + b^2)x + (1 + c^2x) = x + a^2x + x + b^2x + 1 + c^2x = 1 + (a^2+b^2+c^2+2)x$.

Now, we use the given condition: $a^2 + b^2 + c^2 = -2$. Substituting this into the common term $1 + (a^2+b^2+c^2+2)x$: $1 + (-2 + 2)x = 1 + (0)x = 1$.

So, after applying the column operation and using the given condition, the determinant becomes: $f\left( x \right) = \left| {\begin{matrix} {1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \\ {1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x} & {1 + {b^2}x} & {\left( {1 + {c^2}} \right)x} \\ {1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x} & {\left( {1 + {b^2}} \right)x} & {1 + {c^2}x} \\ \end{matrix} } \right|$ $f\left( x \right) = \left| {\begin{matrix} 1 & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \\ 1 & {1 + {b^2}x} & {\left( {1 + {c^2}} \right)x} \\ 1 & {\left( {1 + {b^2}} \right)x} & {1 + {c^2}x} \\ \end{matrix} } \right|$

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Step 2: Further Simplification using Row Operations

Now that the first column consists entirely of '1's, we can easily create zeros using row operations. This is a standard technique to simplify determinants before expansion.

Operations:

1. Apply $R_1 \to R_1 - R_2$.
2. Apply $R_2 \to R_2 - R_3$.

Explanation: These operations subtract the elements of the row below from the current row. This will make the first two entries in the first column zero, significantly simplifying the determinant for expansion.

Let's perform the operations:

• For $R_1 \to R_1 - R_2$:

  • $C_{11} \to 1 - 1 = 0$
  • $C_{12} \to (1 + b^2)x - (1 + b^2x) = x + b^2x - 1 - b^2x = x - 1$
  • $C_{13} \to (1 + c^2)x - (1 + c^2)x = 0$

• For $R_2 \to R_2 - R_3$:

  • $C_{21} \to 1 - 1 = 0$
  • $C_{22} \to (1 + b^2x) - (1 + b^2)x = 1 + b^2x - x - b^2x = 1 - x$
  • $C_{23} \to (1 + c^2)x - (1 + c^2x) = x + c^2x - 1 - c^2x = x - 1$

The determinant now becomes: $f\left( x \right) = \left| {\begin{matrix} 0 & {x - 1} & 0 \\ 0 & {1 - x} & {x - 1} \\ 1 & {\left( {1 + {b^2}} \right)x} & {1 + {c^2}x} \\ \end{matrix} } \right|$

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Step 3: Expand the Determinant

We now have a determinant with two zeros in the first column. This makes expansion straightforward.

Method: Expand the determinant along the first column ($C_1$). Explanation: When expanding a determinant along a row or column, we multiply each element by its cofactor. If an element is zero, its contribution to the sum is zero. Therefore, expanding along a column with many zeros minimizes calculations.

Expanding along $C_1$: $f(x) = 0 \cdot (\text{cofactor of } C_{11}) - 0 \cdot (\text{cofactor of } C_{21}) + 1 \cdot (\text{cofactor of } C_{31})$ $f(x) = 1 \cdot \left| {\begin{matrix} {x - 1} & 0 \\ {1 - x} & {x - 1} \\ \end{matrix} } \right|$ Now, evaluate the $2 \times 2$ determinant: $f(x) = (x - 1)(x - 1) - (0)(1 - x)$ $f(x) = (x - 1)^2 - 0$ $f(x) = (x - 1)^2$

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Step 4: Determine the Degree of the Polynomial

We have found $f(x) = (x - 1)^2$. To find the degree, we expand this expression: $f(x) = (x - 1)^2 = x^2 - 2x + 1$.

Explanation: The degree of a polynomial is the highest power of the variable (in this case, $x$) in its simplified form.

In $f(x) = x^2 - 2x + 1$, the highest power of $x$ is $2$. Therefore, the degree of the polynomial $f(x)$ is $2$.

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Tips and Common Mistakes:

• Strategic Operations: Always look for opportunities to create zeros or common factors using row/column operations. This is the most efficient way to simplify determinants.
• Utilize Given Conditions: The condition $a^2 + b^2 + c^2 = -2$ was crucial. Don't forget to incorporate such conditions at the appropriate step.
• Careful with Arithmetic: Even simple subtractions or additions can lead to errors. Double-check your calculations after each operation.
• Degree Definition: Remember that the degree is the highest power of the variable of interest ($x$ in this case), after the polynomial is fully expanded and simplified. The terms $a^2, b^2, c^2$ are constants for the purpose of determining the polynomial's degree in $x$.

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Conclusion and Key Takeaway:

By systematically applying determinant properties (column operation $C_1 \to C_1 + C_2 + C_3$ followed by row operations $R_1 \to R_1 - R_2$ and $R_2 \to R_2 - R_3$) and utilizing the given condition $a^2 + b^2 + c^2 = -2$, we simplified the complex determinant into a much simpler $2 \times 2$ determinant. This led to $f(x) = (x-1)^2$, which expands to $x^2 - 2x + 1$. The highest power of $x$ in this polynomial is $2$.

Thus, $f(x)$ is a polynomial of degree $2$.

The final answer is $\boxed{D}$.