If `Delta(x)=|{:(x^2+4x-3,2x+4,13),(2x^2+5x-9,4x+5,26),(8x^2-6x+1,16x-6,104):}|=ax^3+bx^2+cx+d` then

Options

A. a=3
B. b=0
C. c=0
D. none of these

Answer

To solve the determinant \( \Delta(x) = \begin{vmatrix} x^2 + 4x - 3 & 2x + 4 & 13 \\ 2x^2 + 5x - 9 & 4x + 5 & 26 \\ 8x^2 - 6x + 1 & 16x - 6 & 104 \end{vmatrix} \), we will perform row operations to simplify the determinant and then expand it.

### Step 1: Write the Determinant
The determinant is given as:
\[ \Delta(x) = \begin{vmatrix} x^2 + 4x - 3 & 2x + 4 & 13 \\ 2x^2 + 5x - 9 & 4x + 5 & 26 \\ 8x^2 - 6x + 1 & 16x - 6 & 104 \end{vmatrix} \]

### Step 2: Perform Row Operations
We will perform the following row operations:
- \( R_2 \rightarrow R_2 - 2R_1 \)
- \( R_3 \rightarrow R_3 - 8R_1 \)

Calculating \( R_2 - 2R_1 \):
\[ R_2 = (2x^2 + 5x - 9) - 2(x^2 + 4x - 3) = 2x^2 + 5x - 9 - 2x^2 - 8x + 6 = -3x - 3 \]

Calculating \( R_3 - 8R_1 \):
\[ R_3 = (8x^2 - 6x + 1) - 8(x^2 + 4x - 3) = 8x^2 - 6x + 1 - 8x^2 - 32x + 24 = -38x + 25 \]

Now, our determinant looks like:
\[ \Delta(x) = \begin{vmatrix} x^2 + 4x - 3 & 2x + 4 & 13 \\ -3x - 3 & -3 & 0 \\ -38x + 25 & 16x - 6 & 0 \end{vmatrix} \]

### Step 3: Expand the Determinant
Now, we can expand the determinant. Since the third column has a zero, we can expand along that column:
\[ \Delta(x) = 13 \cdot \begin{vmatrix} -3x - 3 & -3 \\ -38x + 25 & 16x - 6 \end{vmatrix} \]

Calculating the 2x2 determinant:
\[ = (-3x - 3)(16x - 6) - (-3)(-38x + 25) \]
\[ = (-48x^2 + 18x - 48) - (114x - 75) \]
\[ = -48x^2 + 18x - 48 - 114x + 75 \]
\[ = -48x^2 - 96x + 27 \]

Now, substituting back into the determinant:
\[ \Delta(x) = 13(-48x^2 - 96x + 27) = -624x^2 - 1248x + 351 \]

### Step 4: Identify Coefficients
Now, we can express \( \Delta(x) \) in the form \( ax^3 + bx^2 + cx + d \):
\[ \Delta(x) = 0x^3 - 624x^2 - 1248x + 351 \]
From this, we identify:
- \( a = 0 \)
- \( b = -624 \)
- \( c = -1248 \)
- \( d = 351 \)

### Final Answer
Thus, the values of \( a, b, c, \) and \( d \) are:
- \( a = 0 \)
- \( b = -624 \)
- \( c = -1248 \)
- \( d = 351 \)