Answer: If we divide a polynomial by (x - 2) and obtain a quotient without a remainder, then (x - 2) must be a factor of the polynomial. Therefore, we need to find the factor of (x - 2) that will evenly divide the polynomial x^3 - 4x^2 + 3x - 9.
To do this, we can use long division or synthetic division. Let's use synthetic division:
2 | 1 - 4 + 3 - 9
| 2 - 4 - 2
|---------------
| 1 - 2 - 1 - 11
The result is a quotient of x^2 - 2x - 1 with a remainder of -11.
Since we want to obtain a polynomial that is exactly divisible by (x - 2), we need to subtract the remainder from the original polynomial, which gives:
x^3 - 4x^2 + 3x - 9 - (-11) = x^3 - 4x^2 + 3x + 2
Therefore, we need to subtract 11 from x^3 - 4x^2 + 3x - 9 to obtain a polynomial which is exactly divisible by (x-2). The resulting polynomial is x^3 - 4x^2 + 3x + 2.
Explanation: