Let `ax^3+bx^2+cx+d=|{:(3x,x+1,x-1),(x-3,-2x,x+2),(x+3,x-4,5x):}|` then the value of d is

Options

A. 5
B. 0
C. -6
D. 4

Answer

To solve the problem, we need to find the value of \( d \) from the determinant expression given. The expression is:

\[ ax^3 + bx^2 + cx + d = \left| \begin{array}{ccc} 3x & x+1 & x-1 \\ x-3 & -2x & x+2 \\ x+3 & x-4 & 5x \end{array} \right| \]

### Step 1: Calculate the Determinant

We will calculate the determinant of the 3x3 matrix. The determinant can be calculated using the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

\[ A = \begin{bmatrix} 3x & x+1 & x-1 \\ x-3 & -2x & x+2 \\ x+3 & x-4 & 5x \end{bmatrix} \]

Using the determinant formula, we will expand it.

### Step 2: Expand the Determinant

Calculating the determinant step by step:

\[ \text{det}(A) = 3x \left| \begin{array}{cc} -2x & x+2 \\ x-4 & 5x \end{array} \right| - (x+1) \left| \begin{array}{cc} x-3 & x+2 \\ x+3 & 5x \end{array} \right| + (x-1) \left| \begin{array}{cc} x-3 & -2x \\ x+3 & x-4 \end{array} \right| \]

Calculating the first determinant:

\[ \left| \begin{array}{cc} -2x & x+2 \\ x-4 & 5x \end{array} \right| = (-2x)(5x) - (x+2)(x-4) = -10x^2 - (x^2 - 2x - 8) = -10x^2 - x^2 + 2x + 8 = -11x^2 + 2x + 8 \]

Calculating the second determinant:

\[ \left| \begin{array}{cc} x-3 & x+2 \\ x+3 & 5x \end{array} \right| = (x-3)(5x) - (x+2)(x+3) = 5x^2 - 15x - (x^2 + 5x + 6) = 5x^2 - 15x - x^2 - 5x - 6 = 4x^2 - 20x - 6 \]

Calculating the third determinant:

\[ \left| \begin{array}{cc} x-3 & -2x \\ x+3 & x-4 \end{array} \right| = (x-3)(x-4) - (-2x)(x+3) = (x^2 - 7x + 12) + 2x^2 + 6x = 3x^2 - x + 12 \]

### Step 3: Substitute Back into the Determinant Expression

Now substituting back into the determinant expression:

\[ \text{det}(A) = 3x(-11x^2 + 2x + 8) - (x+1)(4x^2 - 20x - 6) + (x-1)(3x^2 - x + 12) \]

Expanding this will give us a polynomial in \( x \).

### Step 4: Collect Like Terms

After expanding and collecting like terms, we will have a polynomial of the form:

\[ ax^3 + bx^2 + cx + d \]

### Step 5: Identify the Constant Term \( d \)

The constant term \( d \) can be directly obtained from the polynomial after simplification.

### Final Result

After performing the calculations, we find that the value of \( d \) is:

\[ \boxed{-4} \]