The polynomial p(x) is x - 1, and the integer k is 7.
Here's the division of the polynomials:
1. Set up the long division:
x - 2 | x^2 - 3x + 9
2. Divide the first term of the numerator by the first term of the denominator:
x
x - 2 | x^2 - 3x + 9
- (x^2 - 2x)
3. Subtract the result from the numerator and bring down the next term:
x
x - 2 | x^2 - 3x + 9
- (x^2 - 2x)
--------
-x + 9
4. Divide the first term of the new numerator by the first term of the denominator:
x - 1
x - 2 | x^2 - 3x + 9
- (x^2 - 2x)
--------
-x + 9
- (-x + 2)
5. Subtract the result from the numerator:
x - 1
x - 2 | x^2 - 3x + 9
- (x^2 - 2x)
--------
-x + 9
- (-x + 2)
--------
7
6. There are no more terms to bring down.
Therefore, the answer is:
p(x) = x - 1 + (7)/(x - 2)
So, the polynomial p(x) is x - 1, and the integer k is 7.