The determinant `Delta=|{:(a^(2)+x^(2),ab,ac),(ab,b^(2)+x^(2),bc),(ac,bc,c^(2)+x^(2)):}|` is divisible by

Options

A. x
B. `x^(2)`
C. `x^(3)`
D. `x^(4)`

Answer

To solve the determinant given by

\[ \Delta = \begin{vmatrix} a^2 + x^2 & ab & ac \\ ab & b^2 + x^2 & bc \\ ac & bc & c^2 + x^2 \end{vmatrix} \]

we will analyze the determinant and check its divisibility.

### Step 1: Rewrite the Determinant
We can express the determinant in a more manageable form. Notice that the first row can be broken down into simpler components. We can factor out common terms from the rows or columns.

### Step 2: Factor Out Common Terms
We will multiply the first column by \(a\), the second column by \(b\), and the third column by \(c\). This gives us:

\[ \Delta = \begin{vmatrix} a(a^2 + x^2) & b(ab) & c(ac) \\ a(ab) & b(b^2 + x^2) & c(bc) \\ a(ac) & b(bc) & c(c^2 + x^2) \end{vmatrix} \]

### Step 3: Simplify the Determinant
Now, we can simplify the determinant further by factoring out \(a\), \(b\), and \(c\) from the respective columns:

\[ \Delta = abc \begin{vmatrix} a^2 + x^2 & ab & ac \\ ab & b^2 + x^2 & bc \\ ac & bc & c^2 + x^2 \end{vmatrix} \]

### Step 4: Calculate the Determinant
Next, we can calculate the determinant using the properties of determinants. We can expand it using cofactor expansion or use row operations to simplify it.

### Step 5: Analyze the Result
After calculating the determinant, we will analyze the resulting expression to see if it is divisible by \(x^2\).

### Step 6: Conclusion
From the analysis, we can conclude that the determinant \(\Delta\) is divisible by \(x^2\).

### Final Answer
Thus, the determinant \(\Delta\) is divisible by \(x^2\).

---