If `f(x) = ax^(6) +bx^(5) +cx^(4) +dx^(3) +ex^(2)+fx+g`
`=|(x^(2)-2x+3,7x+2,x+4),(2x+7,x^2-x+2,3x),(3,2x-1,x^2-4x+7)|` then g =

Options

A. `-200`
B. `100`
C. `112`
D. `-108`

Answer

To solve the problem, we need to find the value of \( g \) in the polynomial \( f(x) = ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g \) given that it is equal to the determinant:

\[ \begin{vmatrix} x^2 - 2x + 3 & 7x + 2 & x + 4 \\ 2x + 7 & x^2 - x + 2 & 3x \\ 3 & 2x - 1 & x^2 - 4x + 7 \end{vmatrix} \]

### Step 1: Evaluate \( f(0) \)

To find \( g \), we can evaluate \( f(0) \):

\[ f(0) = g \]

### Step 2: Substitute \( x = 0 \) into the determinant

Now, we substitute \( x = 0 \) into the determinant:

\[ \begin{vmatrix} 0^2 - 2(0) + 3 & 7(0) + 2 & 0 + 4 \\ 2(0) + 7 & 0^2 - 0 + 2 & 3(0) \\ 3 & 2(0) - 1 & 0^2 - 4(0) + 7 \end{vmatrix} \]

This simplifies to:

\[ \begin{vmatrix} 3 & 2 & 4 \\ 7 & 2 & 0 \\ 3 & -1 & 7 \end{vmatrix} \]

### Step 3: Calculate the determinant

Now we calculate the determinant using the formula for a 3x3 matrix:

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

For our matrix:

- \( a = 3, b = 2, c = 4 \)
- \( d = 7, e = 2, f = 0 \)
- \( g = 3, h = -1, i = 7 \)

Calculating the determinant:

\[ \text{det} = 3(2 \cdot 7 - 0 \cdot (-1)) - 2(7 \cdot 7 - 0 \cdot 3) + 4(7 \cdot (-1) - 2 \cdot 3) \]

Calculating each term:

1. \( 3(14 - 0) = 42 \)
2. \( -2(49 - 0) = -98 \)
3. \( 4(-7 - 6) = 4(-13) = -52 \)

Now combine these results:

\[ \text{det} = 42 - 98 - 52 = 42 - 150 = -108 \]

### Conclusion

Thus, we find that:

\[ g = -108 \]