If the expression `|{:(x^2+x+3,1,4),(2x^4+x^3+2x+1,2,3),(x^2+x,1,1):}|` is equal to `ax^4+bx^3+cx^2+dx+e` , then the value of e is equal to

Options

A. zero
B. 1
C. 2
D. none of these

Answer

To solve the determinant \( | \begin{pmatrix} x^2 + x + 3 & 1 & 4 \\ 2x^4 + x^3 + 2x + 1 & 2 & 3 \\ x^2 + x & 1 & 1 \end{pmatrix} | \) and find the constant term \( e \) in the expression \( ax^4 + bx^3 + cx^2 + dx + e \), we will proceed step by step.

### Step 1: Write the determinant
We start with the determinant:

\[ D = \begin{vmatrix} x^2 + x + 3 & 1 & 4 \\ 2x^4 + x^3 + 2x + 1 & 2 & 3 \\ x^2 + x & 1 & 1 \end{vmatrix} \]

### Step 2: Expand the determinant
We will use the first row to expand the determinant:

\[ D = (x^2 + x + 3) \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} - 1 \begin{vmatrix} 2x^4 + x^3 + 2x + 1 & 3 \\ x^2 + x & 1 \end{vmatrix} + 4 \begin{vmatrix} 2x^4 + x^3 + 2x + 1 & 2 \\ x^2 + x & 1 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
1. For the first determinant:

\[ \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} = (2 \cdot 1) - (3 \cdot 1) = 2 - 3 = -1 \]

2. For the second determinant:

\[ \begin{vmatrix} 2x^4 + x^3 + 2x + 1 & 3 \\ x^2 + x & 1 \end{vmatrix} = (2x^4 + x^3 + 2x + 1) \cdot 1 - (3 \cdot (x^2 + x)) = 2x^4 + x^3 + 2x + 1 - 3x^2 - 3x = 2x^4 + x^3 - 3x^2 - x + 1 \]

3. For the third determinant:

\[ \begin{vmatrix} 2x^4 + x^3 + 2x + 1 & 2 \\ x^2 + x & 1 \end{vmatrix} = (2x^4 + x^3 + 2x + 1) \cdot 1 - (2 \cdot (x^2 + x)) = 2x^4 + x^3 + 2x + 1 - 2x^2 - 2x = 2x^4 + x^3 - 2x^2 + 1 \]

### Step 4: Substitute back into the determinant
Now substituting back into the determinant expression:

\[ D = (x^2 + x + 3)(-1) - 1(2x^4 + x^3 - 3x^2 - x + 1) + 4(2x^4 + x^3 - 2x^2 + 1) \]

### Step 5: Simplify the expression
1. The first term becomes:

\[ -(x^2 + x + 3) = -x^2 - x - 3 \]

2. The second term becomes:

\[ -(2x^4 + x^3 - 3x^2 - x + 1) = -2x^4 - x^3 + 3x^2 + x - 1 \]

3. The third term becomes:

\[ 4(2x^4 + x^3 - 2x^2 + 1) = 8x^4 + 4x^3 - 8x^2 + 4 \]

### Step 6: Combine all terms
Combining all the terms:

\[ D = (-x^2 - x - 3) + (-2x^4 - x^3 + 3x^2 + x - 1) + (8x^4 + 4x^3 - 8x^2 + 4) \]

Combining like terms:

- \(x^4\) terms: \(-2x^4 + 8x^4 = 6x^4\)
- \(x^3\) terms: \(-x^3 + 4x^3 = 3x^3\)
- \(x^2\) terms: \(-x^2 + 3x^2 - 8x^2 = -6x^2\)
- \(x\) terms: \(-x + x = 0\)
- Constant terms: \(-3 - 1 + 4 = 0\)

### Final Result
Thus, the determinant simplifies to:

\[ D = 6x^4 + 3x^3 - 6x^2 + 0 + 0 \]

The constant term \( e \) is:

\[ e = 0 \]