If the polynomial `x^(4)-6x^(3)+16x^(2)-25x+10` is divided by another polynomial `x^(2)-2x+k`, the remainder comes out to be (x-a). Find k and a.

Explanation

To solve the problem, we need to find the values of \( k \) and \( a \) such that when the polynomial \( P(x) = x^4 - 6x^3 + 16x^2 - 25x + 10 \) is divided by \( D(x) = x^2 - 2x + k \), the remainder is \( R(x) = x - a \).

### Step-by-Step Solution:

1. **Set up the division**: We need to divide \( P(x) \) by \( D(x) \) and express \( P(x) \) in the form:
\[ P(x) = D(x) \cdot Q(x) + R(x) \]
where \( R(x) = x - a \).

2. **Perform polynomial long division**:
- Divide the leading term of \( P(x) \) (which is \( x^4 \)) by the leading term of \( D(x) \) (which is \( x^2 \)). This gives us \( x^2 \).
- Multiply \( D(x) \) by \( x^2 \):
\[ x^2 \cdot (x^2 - 2x + k) = x^4 - 2x^3 + kx^2 \]
- Subtract this from \( P(x) \):
\[ P(x) - (x^4 - 2x^3 + kx^2) = (0 + 2x^3 + (16 - k)x^2 - 25x + 10) \]

3. **Continue the division**:
- Now, divide the leading term \( 2x^3 \) by \( x^2 \) to get \( 2x \).
- Multiply \( D(x) \) by \( 2x \):
\[ 2x \cdot (x^2 - 2x + k) = 2x^3 - 4x^2 + 2kx \]
- Subtract this from the previous result:
\[ (2x^3 + (16 - k)x^2 - 25x + 10) - (2x^3 - 4x^2 + 2kx) = (20 - k)x^2 + (-25 - 2k)x + 10 \]

4. **Final division step**:
- Divide \( (20 - k)x^2 \) by \( x^2 \) to get \( 20 - k \).
- Multiply \( D(x) \) by \( 20 - k \):
\[ (20 - k)(x^2 - 2x + k) = (20 - k)x^2 - 2(20 - k)x + (20 - k)k \]
- Subtract this from the previous result:
\[ ((20 - k)x^2 + (-25 - 2k)x + 10) - ((20 - k)x^2 - 2(20 - k)x + (20 - k)k) = (2(20 - k) - 25 - 2k)x + (10 - (20 - k)k) \]

5. **Set the remainder equal to \( R(x) \)**:
- From the division, we have:
\[ (2(20 - k) - 25 - 2k)x + (10 - (20 - k)k) = x - a \]
- This gives us two equations:
1. \( 2(20 - k) - 25 - 2k = 1 \)
2. \( 10 - (20 - k)k = -a \)

6. **Solve for \( k \)**:
- From the first equation:
\[ 40 - 2k - 25 - 2k = 1 \implies 15 - 4k = 1 \implies 4k = 14 \implies k = \frac{14}{4} = 3.5 \]

7. **Substitute \( k \) back to find \( a \)**:
- Substitute \( k = 3.5 \) into the second equation:
\[ 10 - (20 - 3.5)(3.5) = -a \implies 10 - (16.5)(3.5) = -a \]
\[ 10 - 57.75 = -a \implies -47.75 = -a \implies a = 47.75 \]

### Final Values:
- \( k = 3.5 \)
- \( a = 47.75 \)