If `(x-1)^(2)` is a factor of `f (x)=x^(3)+bx+c` , then find the remainder when `f (x)` is divided by `(x-2)`.

Options

A. 2
B. `-3`
C. 4
D. `-4`

Explanation

To solve the problem, we need to find the remainder when \( f(x) = x^3 + bx + c \) is divided by \( x - 2 \), given that \( (x - 1)^2 \) is a factor of \( f(x) \).

### Step 1: Use the Factor Theorem
Since \( (x - 1)^2 \) is a factor of \( f(x) \), it implies that:
1. \( f(1) = 0 \)
2. \( f'(1) = 0 \)

### Step 2: Calculate \( f(1) \)
Substituting \( x = 1 \) into \( f(x) \):
\[ f(1) = 1^3 + b(1) + c = 1 + b + c = 0 \]
This gives us our first equation:
\[ b + c = -1 \quad \text{(Equation 1)} \]

### Step 3: Differentiate \( f(x) \)
Now, we differentiate \( f(x) \):
\[ f'(x) = 3x^2 + b \]

### Step 4: Calculate \( f'(1) \)
Substituting \( x = 1 \) into \( f'(x) \):
\[ f'(1) = 3(1)^2 + b = 3 + b = 0 \]
This gives us our second equation:
\[ b = -3 \quad \text{(Equation 2)} \]

### Step 5: Substitute \( b \) into Equation 1
Now, substitute \( b = -3 \) into Equation 1:
\[ -3 + c = -1 \]
Solving for \( c \):
\[ c = -1 + 3 = 2 \quad \text{(Equation 3)} \]

### Step 6: Write the polynomial \( f(x) \)
Now we have \( b \) and \( c \):
\[ b = -3, \quad c = 2 \]
Thus, the polynomial becomes:
\[ f(x) = x^3 - 3x + 2 \]

### Step 7: Find the remainder when \( f(x) \) is divided by \( x - 2 \)
Using the Remainder Theorem, the remainder when \( f(x) \) is divided by \( x - 2 \) is \( f(2) \):
\[ f(2) = 2^3 - 3(2) + 2 \]
Calculating this:
\[ f(2) = 8 - 6 + 2 = 4 \]

### Conclusion
The remainder when \( f(x) \) is divided by \( x - 2 \) is:
\[ \boxed{4} \]