If the equation of the plane passing through the point (4, 5, 1), (3, 9, 4), (-4, 4, 4) is `ax + by + cz+d= 0` then `a^(4)-b^(3)+c^(2)+d` is equal to _____________.

Answer

To find the value of \( a^4 - b^3 + c^2 + d \) for the plane passing through the points \( (4, 5, 1) \), \( (3, 9, 4) \), and \( (-4, 4, 4) \), we will follow these steps:

### Step 1: Use the formula for the equation of a plane through three points
The general form of the equation of a plane given three points \( (x_1, y_1, z_1) \), \( (x_2, y_2, z_2) \), and \( (x_3, y_3, z_3) \) is given by the determinant:

\[ \begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0 \]

### Step 2: Substitute the given points into the formula
Let \( (x_1, y_1, z_1) = (4, 5, 1) \), \( (x_2, y_2, z_2) = (3, 9, 4) \), and \( (x_3, y_3, z_3) = (-4, 4, 4) \).

Calculating the differences:
- \( x_2 - x_1 = 3 - 4 = -1 \)
- \( y_2 - y_1 = 9 - 5 = 4 \)
- \( z_2 - z_1 = 4 - 1 = 3 \)

- \( x_3 - x_1 = -4 - 4 = -8 \)
- \( y_3 - y_1 = 4 - 5 = -1 \)
- \( z_3 - z_1 = 4 - 1 = 3 \)

Now, we can write the determinant:

\[ \begin{vmatrix} x - 4 & y - 5 & z - 1 \\ -1 & 4 & 3 \\ -8 & -1 & 3 \end{vmatrix} = 0 \]

### Step 3: Calculate the determinant
Expanding the determinant:

\[ (x - 4) \begin{vmatrix} 4 & 3 \\ -1 & 3 \end{vmatrix} - (y - 5) \begin{vmatrix} -1 & 3 \\ -8 & 3 \end{vmatrix} + (z - 1) \begin{vmatrix} -1 & 4 \\ -8 & -1 \end{vmatrix} = 0 \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 4 & 3 \\ -1 & 3 \end{vmatrix} = (4)(3) - (3)(-1) = 12 + 3 = 15 \)
2. \( \begin{vmatrix} -1 & 3 \\ -8 & 3 \end{vmatrix} = (-1)(3) - (3)(-8) = -3 + 24 = 21 \)
3. \( \begin{vmatrix} -1 & 4 \\ -8 & -1 \end{vmatrix} = (-1)(-1) - (4)(-8) = 1 + 32 = 33 \)

Substituting back into the equation:

\[ (x - 4)(15) - (y - 5)(21) + (z - 1)(33) = 0 \]

### Step 4: Expand and simplify
Expanding this gives:

\[ 15x - 60 - 21y + 105 + 33z - 33 = 0 \]

Combining like terms:

\[ 15x - 21y + 33z + 12 = 0 \]

### Step 5: Rearranging to standard form
Rearranging gives us:

\[ 15x - 21y + 33z + 12 = 0 \]

From this, we can identify the coefficients:
- \( a = 15 \)
- \( b = -21 \)
- \( c = 33 \)
- \( d = 12 \)

### Step 6: Calculate \( a^4 - b^3 + c^2 + d \)
Now we compute:

\[ a^4 = 15^4 = 50625 \]
\[ b^3 = (-21)^3 = -9261 \]
\[ c^2 = 33^2 = 1089 \]
\[ d = 12 \]

Putting it all together:

\[ a^4 - b^3 + c^2 + d = 50625 - (-9261) + 1089 + 12 \]

Calculating this step-by-step:

1. \( 50625 + 9261 = 59886 \)
2. \( 59886 + 1089 = 60975 \)
3. \( 60975 + 12 = 60987 \)

Thus, the final answer is:

\[ \boxed{60987} \]