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 given in the question, we will follow the steps below:

Given 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 1: Identify the constant terms in each row
We need to extract the constant terms from each polynomial in the first column of the determinant.

- From the first row, the constant term is **3**.
- From the second row, the constant term is **1**.
- From the third row, the constant term is **0** (since \(x^2 + x\) has no constant term).

### Step 2: Set up the determinant with constant terms
Now we can simplify the determinant by focusing on the constant terms:

\[ D = \begin{vmatrix} 3 & 1 & 4 \\ 1 & 2 & 3 \\ 0 & 1 & 1 \end{vmatrix} \]

### Step 3: Calculate the determinant
We will calculate the determinant using the formula for a \(3 \times 3\) determinant:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where \(a, b, c\) are the elements of the first row, and \(d, e, f, g, h, i\) are the elements of the second and third rows.

Substituting the values:

\[ D = 3 \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} - 1 \begin{vmatrix} 1 & 3 \\ 0 & 1 \end{vmatrix} + 4 \begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix} \]

Calculating each of the \(2 \times 2\) determinants:

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

2. \(\begin{vmatrix} 1 & 3 \\ 0 & 1 \end{vmatrix} = (1 \cdot 1 - 3 \cdot 0) = 1 - 0 = 1\)

3. \(\begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix} = (1 \cdot 1 - 2 \cdot 0) = 1 - 0 = 1\)

Now substituting these back into the determinant calculation:

\[ D = 3(-1) - 1(1) + 4(1) \]
\[ D = -3 - 1 + 4 \]
\[ D = 0 \]

### Conclusion
Since the determinant \(D\) is equal to \(0\), and we are given that \(D = ax^4 + bx^3 + cx^2 + dx + e\), we can conclude that the constant term \(e\) is equal to \(0\).

Thus, the value of \(e\) is:

\[ \boxed{0} \]