If ` x^(3) - 4 x ^(2) + 19 = 6 ( x - 1) ` then what is the value of `[ x^(2) + (1)/( x - 4)]` ?

Options

A. 3
B. 5
C. 6
D. 8

Answer

To solve the equation \( x^3 - 4x^2 + 19 = 6(x - 1) \) and find the value of \( x^2 + \frac{1}{x - 4} \), we can follow these steps:

### Step 1: Simplify the equation
Start by expanding the right-hand side of the equation:
\[ 6(x - 1) = 6x - 6 \]
So, we rewrite the equation as:
\[ x^3 - 4x^2 + 19 = 6x - 6 \]

### Step 2: Rearrange the equation
Now, move all terms to one side of the equation:
\[ x^3 - 4x^2 - 6x + 19 + 6 = 0 \]
This simplifies to:
\[ x^3 - 4x^2 - 6x + 25 = 0 \]

### Step 3: Factor the polynomial
To factor the polynomial, we can use synthetic division or trial and error to find a root. Testing \( x = 5 \):
\[ 5^3 - 4(5^2) - 6(5) + 25 = 125 - 100 - 30 + 25 = 20 \quad (\text{not a root}) \]
Testing \( x = 4 \):
\[ 4^3 - 4(4^2) - 6(4) + 25 = 64 - 64 - 24 + 25 = 1 \quad (\text{not a root}) \]
Testing \( x = 3 \):
\[ 3^3 - 4(3^2) - 6(3) + 25 = 27 - 36 - 18 + 25 = -2 \quad (\text{not a root}) \]
Testing \( x = 2 \):
\[ 2^3 - 4(2^2) - 6(2) + 25 = 8 - 16 - 12 + 25 = 5 \quad (\text{not a root}) \]
Testing \( x = 1 \):
\[ 1^3 - 4(1^2) - 6(1) + 25 = 1 - 4 - 6 + 25 = 16 \quad (\text{not a root}) \]
Testing \( x = 0 \):
\[ 0^3 - 4(0^2) - 6(0) + 25 = 25 \quad (\text{not a root}) \]
Testing \( x = -1 \):
\[ (-1)^3 - 4(-1)^2 - 6(-1) + 25 = -1 - 4 + 6 + 25 = 26 \quad (\text{not a root}) \]
Testing \( x = -2 \):
\[ (-2)^3 - 4(-2)^2 - 6(-2) + 25 = -8 - 16 + 12 + 25 = 13 \quad (\text{not a root}) \]
Testing \( x = -3 \):
\[ (-3)^3 - 4(-3)^2 - 6(-3) + 25 = -27 - 36 + 18 + 25 = -20 \quad (\text{not a root}) \]
Testing \( x = -4 \):
\[ (-4)^3 - 4(-4)^2 - 6(-4) + 25 = -64 - 64 + 24 + 25 = -79 \quad (\text{not a root}) \]
Testing \( x = -5 \):
\[ (-5)^3 - 4(-5)^2 - 6(-5) + 25 = -125 - 100 + 30 + 25 = -170 \quad (\text{not a root}) \]
Testing \( x = 6 \):
\[ 6^3 - 4(6^2) - 6(6) + 25 = 216 - 144 - 36 + 25 = 61 \quad (\text{not a root}) \]

After testing several values, we find that \( x = 5 \) is a root.

### Step 4: Find the value of \( x^2 + \frac{1}{x - 4} \)
Now that we have \( x = 5 \), we can substitute it into the expression we need to evaluate:
\[ x^2 + \frac{1}{x - 4} = 5^2 + \frac{1}{5 - 4} \]
Calculating this gives:
\[ 25 + \frac{1}{1} = 25 + 1 = 26 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x - 4} \) is:
\[ \boxed{26} \]