What should be added to the polynomial `x^(3)-3x^(2)+6x-15`, so that it is completely divisible by `x-3` ?

Answer

To determine what should be added to the polynomial \( P(x) = x^3 - 3x^2 + 6x - 15 \) so that it is completely divisible by \( x - 3 \), we can follow these steps:

### Step 1: Find the remainder when dividing \( P(x) \) by \( x - 3 \)
To find the remainder, we can use the Remainder Theorem, which states that the remainder of the division of a polynomial \( P(x) \) by \( x - c \) is \( P(c) \). Here, \( c = 3 \).

\[ P(3) = (3)^3 - 3(3)^2 + 6(3) - 15 \]

### Step 2: Calculate \( P(3) \)
Now, we calculate \( P(3) \):

\[ P(3) = 27 - 27 + 18 - 15 \]
\[ P(3) = 27 - 27 + 18 - 15 = 0 + 18 - 15 = 3 \]

### Step 3: Determine what to add to make the polynomial divisible
Since the remainder \( P(3) = 3 \), to make the polynomial \( P(x) \) completely divisible by \( x - 3 \), we need to subtract this remainder from the polynomial. Therefore, we need to add \( -3 \) to \( P(x) \).

### Conclusion
Thus, the value that should be added to the polynomial \( x^3 - 3x^2 + 6x - 15 \) so that it is completely divisible by \( x - 3 \) is:

\[ \boxed{-3} \]