If 5 and 7 are the roots of the equation `|{:(x,4,5),(7,x,7),(5,8,x):}|=0`, then what is the third root ?

Options

A. -12
B. 9
C. 13
D. 14

Explanation

To find the third root of the equation given by the determinant \( |{:(x,4,5),(7,x,7),(5,8,x):}|=0 \), where 5 and 7 are known roots, we will follow these steps:

### Step 1: Set Up the Determinant
The determinant can be expressed as:
\[ D = \begin{vmatrix} x & 4 & 5 \\ 7 & x & 7 \\ 5 & 8 & x \end{vmatrix} \]

### Step 2: Expand the Determinant
We will expand the determinant using the first row:
\[ D = x \begin{vmatrix} x & 7 \\ 8 & x \end{vmatrix} - 4 \begin{vmatrix} 7 & 7 \\ 5 & x \end{vmatrix} + 5 \begin{vmatrix} 7 & x \\ 5 & 8 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. \( \begin{vmatrix} x & 7 \\ 8 & x \end{vmatrix} = x^2 - 56 \)
2. \( \begin{vmatrix} 7 & 7 \\ 5 & x \end{vmatrix} = 7x - 35 \)
3. \( \begin{vmatrix} 7 & x \\ 5 & 8 \end{vmatrix} = 56 - 5x \)

Substituting these back into our determinant:
\[ D = x(x^2 - 56) - 4(7x - 35) + 5(56 - 5x) \]

### Step 3: Simplify the Expression
Expanding and simplifying:
\[ D = x^3 - 56x - 28x + 140 + 280 - 25x \]
Combining like terms:
\[ D = x^3 - 109x + 420 \]

### Step 4: Factor the Polynomial
Given that 5 and 7 are roots, we can express the polynomial as:
\[ D = (x - 5)(x - 7)(x - r) \]
To find the third root \( r \), we can use polynomial division.

### Step 5: Perform Polynomial Division
Dividing \( x^3 - 109x + 420 \) by \( (x - 5)(x - 7) \):
1. First, calculate \( (x - 5)(x - 7) = x^2 - 12x + 35 \).
2. Now perform the division of \( x^3 - 109x + 420 \) by \( x^2 - 12x + 35 \).

Using long division:
- Divide \( x^3 \) by \( x^2 \) to get \( x \).
- Multiply \( x \) by \( x^2 - 12x + 35 \) to get \( x^3 - 12x^2 + 35x \).
- Subtract this from the original polynomial to get \( 12x^2 - 144x + 420 \).
- Divide \( 12x^2 \) by \( x^2 \) to get \( 12 \).
- Multiply \( 12 \) by \( x^2 - 12x + 35 \) to get \( 12x^2 - 144x + 420 \).
- Subtracting gives a remainder of 0.

### Step 6: Identify the Third Root
The quotient from the division is \( x + 12 \). Therefore, the third root is:
\[ x + 12 = 0 \implies x = -12 \]

### Final Answer
The third root is \( \boxed{-12} \).