Find the remainder when `(x^(3)-4x^(2)+5x+6)` is divided by (x-2).

Options

A. `-8`
B. `8`
C. `2`
D. None of these

Answer

To find the remainder when the polynomial \( P(x) = x^3 - 4x^2 + 5x + 6 \) is divided by \( x - 2 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( P(x) \) by \( x - c \) is equal to \( P(c) \).

### Step-by-Step Solution:

1. **Identify the polynomial and the divisor**:
- The polynomial is \( P(x) = x^3 - 4x^2 + 5x + 6 \).
- The divisor is \( x - 2 \), which means \( c = 2 \).

2. **Evaluate the polynomial at \( x = 2 \)**:
- We need to calculate \( P(2) \):
\[ P(2) = (2)^3 - 4(2)^2 + 5(2) + 6 \]

3. **Calculate each term**:
- Calculate \( (2)^3 = 8 \).
- Calculate \( -4(2)^2 = -4 \times 4 = -16 \).
- Calculate \( 5(2) = 10 \).
- The constant term is \( 6 \).

4. **Combine the results**:
- Now, substitute these values back into the expression:
\[ P(2) = 8 - 16 + 10 + 6 \]

5. **Simplify the expression**:
- Combine the terms:
\[ P(2) = 8 - 16 = -8 \]
\[ -8 + 10 = 2 \]
\[ 2 + 6 = 8 \]

6. **Conclusion**:
- The remainder when \( P(x) \) is divided by \( x - 2 \) is \( 8 \).

### Final Answer:
The remainder is \( 8 \).

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