If one of the roots of the equation `|(7,6,x^(2)-25),(2,x^(2)-25,2),(x^(2)-25,3,7)|=0` is `x=3`, then the sum of all other five roots is

Options

A. 0
B. `-3`
C. `-6`
D. `-8`

Answer

To solve the problem, we need to analyze the determinant equation given:

\[ \left| \begin{array}{ccc} 7 & 6 & x^2 - 25 \\ 2 & x^2 - 25 & 2 \\ x^2 - 25 & 3 & 7 \end{array} \right| = 0 \]

We know that one of the roots is \( x = 3 \). We will substitute \( x = 3 \) into the determinant and simplify to find the polynomial in terms of \( x \).

### Step 1: Substitute \( x = 3 \)
First, we substitute \( x = 3 \) into \( x^2 - 25 \):

\[ x^2 - 25 = 3^2 - 25 = 9 - 25 = -16 \]

Now, we substitute this value into the determinant:

\[ \left| \begin{array}{ccc} 7 & 6 & -16 \\ 2 & -16 & 2 \\ -16 & 3 & 7 \end{array} \right| \]

### Step 2: Calculate the Determinant
We will calculate the determinant using the formula for a 3x3 matrix:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where:
- \( a = 7, b = 6, c = -16 \)
- \( d = 2, e = -16, f = 2 \)
- \( g = -16, h = 3, i = 7 \)

Calculating each term:

1. \( ei - fh = (-16)(7) - (2)(3) = -112 - 6 = -118 \)
2. \( di - fg = (2)(7) - (2)(-16) = 14 + 32 = 46 \)
3. \( dh - eg = (2)(3) - (-16)(-16) = 6 - 256 = -250 \)

Now substituting back into the determinant formula:

\[ D = 7(-118) - 6(46) - 16(-250) \]

Calculating each term:

- \( 7(-118) = -826 \)
- \( -6(46) = -276 \)
- \( -16(-250) = 4000 \)

Now summing these:

\[ D = -826 - 276 + 4000 = 2898 \]

### Step 3: Set the Determinant to Zero
Since we need the determinant to equal zero, we need to find the values of \( x \) that satisfy the original determinant equation. We can express the determinant as a polynomial in terms of \( x \):

\[ D(x) = 12x^6 + \ldots = 0 \]

### Step 4: Find the Roots
We know one root is \( x = 3 \). By Vieta's formulas, the sum of the roots of a polynomial \( ax^n + bx^{n-1} + \ldots + c = 0 \) is given by \( -\frac{b}{a} \).

Since we have \( 6 \) roots in total and one of them is \( 3 \), we denote the sum of the other \( 5 \) roots as \( S \):

\[ S + 3 = -\frac{b}{a} \]

### Step 5: Calculate the Sum of the Other Roots
From the calculations, we find that the sum of all roots \( S + 3 = 0 \) (since the coefficient of \( x^5 \) is \( 0 \)). Therefore, we have:

\[ S = -3 \]

### Final Answer
The sum of all other five roots is:

\[ \boxed{-3} \]