The value of k so that ` x^4 -4x^3 +5x^2 -2x +k` is divisible by ` x^2 -2x +2 ` is

Options

A. 0
B. `-2`
C. `-1`
D. 2

Answer

To find the value of \( k \) such that the polynomial \( P(x) = x^4 - 4x^3 + 5x^2 - 2x + k \) is divisible by \( Q(x) = x^2 - 2x + 2 \), we can use the fact that if \( P(x) \) is divisible by \( Q(x) \), then the remainder when \( P(x) \) is divided by \( Q(x) \) must be zero.

### Step-by-Step Solution:

1. **Set Up the Division**: We need to perform polynomial long division of \( P(x) \) by \( Q(x) \).

2. **Perform Polynomial Long Division**:
- Divide the leading term of \( P(x) \) by the leading term of \( Q(x) \):
\[ \frac{x^4}{x^2} = x^2 \]
- Multiply \( Q(x) \) by \( x^2 \):
\[ x^2(x^2 - 2x + 2) = x^4 - 2x^3 + 2x^2 \]
- Subtract this from \( P(x) \):
\[ (x^4 - 4x^3 + 5x^2) - (x^4 - 2x^3 + 2x^2) = -2x^3 + 3x^2 \]

3. **Continue the Division**:
- Now, bring down the next term from \( P(x) \), which is \( -2x \):
\[ -2x^3 + 3x^2 - 2x \]
- Divide the leading term by the leading term of \( Q(x) \):
\[ \frac{-2x^3}{x^2} = -2x \]
- Multiply \( Q(x) \) by \( -2x \):
\[ -2x(x^2 - 2x + 2) = -2x^3 + 4x^2 - 4x \]
- Subtract this from the current polynomial:
\[ (-2x^3 + 3x^2 - 2x) - (-2x^3 + 4x^2 - 4x) = -x^2 + 2x \]

4. **Final Step in Division**:
- Bring down \( k \):
\[ -x^2 + 2x + k \]
- Divide the leading term:
\[ \frac{-x^2}{x^2} = -1 \]
- Multiply \( Q(x) \) by \( -1 \):
\[ -1(x^2 - 2x + 2) = -x^2 + 2x - 2 \]
- Subtract this from the current polynomial:
\[ (-x^2 + 2x + k) - (-x^2 + 2x - 2) = k + 2 \]

5. **Set the Remainder to Zero**: For \( P(x) \) to be divisible by \( Q(x) \), the remainder must be zero:
\[ k + 2 = 0 \]
Therefore,
\[ k = -2 \]

### Conclusion:
The value of \( k \) such that \( P(x) \) is divisible by \( Q(x) \) is \( k = -2 \).