If `|(2+x,x,x^(2)),(x,2+x,x^(2)),(x^(2),x,2+x)|=(1)/(6)(x-a)(x-b)(x-c)(x-d)` an identity in x where a, b, c, d are independent of x, then the value of `(13)/(25)abcd` is

Answer

To solve the problem, we need to evaluate the determinant and set it equal to the given expression. Let's break it down step by step.

### Step 1: Write down the determinant
We have the determinant:
\[ D = \begin{vmatrix} 2+x & x & x^2 \\ x & 2+x & x^2 \\ x^2 & x & 2+x \end{vmatrix} \]

### Step 2: Calculate the determinant
We can calculate the determinant using the formula for a 3x3 matrix:
\[ D = a(ei-fh) - b(di-fg) + c(dh-eg) \]
where \( a, b, c, d, e, f, g, h, i \) are the elements of the matrix.

Substituting the elements from our matrix:
\[ D = (2+x)((2+x)(2+x) - x^2) - x(x^2(2+x) - x^2) + x^2(x^2x - x(2+x)) \]

Calculating each term:
1. The first term:
\[ (2+x)((2+x)(2+x) - x^2) = (2+x)(4 + 4x + x^2 - x^2) = (2+x)(4 + 4x) = 8 + 8x + 4x + 4x^2 = 8 + 12x + 4x^2 \]

2. The second term:
\[ -x(x^2(2+x) - x^2) = -x(x^2(2+x-1)) = -x(x^2(1+x)) = -x^3 - x^4 \]

3. The third term:
\[ x^2(x^2x - x(2+x)) = x^2(x^3 - 2x - x^2) = x^2(x^3 - x^2 - 2x) = x^5 - x^4 - 2x^3 \]

Combining all the terms:
\[ D = (8 + 12x + 4x^2) - (x^3 + x^4) + (x^5 - x^4 - 2x^3) \]
\[ D = 8 + 12x + 4x^2 - x^4 - 3x^3 + x^5 \]

### Step 3: Set the determinant equal to the given expression
We know:
\[ D = \frac{1}{6}(x-a)(x-b)(x-c)(x-d) \]
This means:
\[ x^5 - 3x^3 - x^4 + 4x^2 + 8 = \frac{1}{6}(x^4 - (a+b+c+d)x^3 + (ab+ac+ad+bc+bd+cd)x^2 - (abc+abd+acd+bcd)x + abcd) \]

### Step 4: Compare coefficients
From the equation, we can compare coefficients of \(x^5, x^4, x^3, x^2, x^1, x^0\) on both sides to find \(a, b, c, d\).

1. Coefficient of \(x^5\): \(1 = \frac{1}{6}\) (no \(x^5\) term on RHS)
2. Coefficient of \(x^4\): \(-1 = -\frac{1}{6}(a+b+c+d)\)
\[ a+b+c+d = 6 \]
3. Coefficient of \(x^3\): \(-3 = \frac{1}{6}(ab+ac+ad+bc+bd+cd)\)
\[ ab+ac+ad+bc+bd+cd = -18 \]
4. Coefficient of \(x^2\): \(4 = \frac{1}{6}(-abc-abd-acd-bcd)\)
\[ abc+abd+acd+bcd = -24 \]
5. Coefficient of \(x^0\): \(8 = \frac{1}{6}(abcd)\)
\[ abcd = 48 \]

### Step 5: Calculate \( \frac{13}{25} abcd \)
Now we can find:
\[ \frac{13}{25} abcd = \frac{13}{25} \times 48 = \frac{624}{25} = 24.96 \]

### Final Answer
Thus, the value of \( \frac{13}{25} abcd \) is:
\[ \boxed{24.96} \]