What number should be added to `x^(3)-9x^(2)-2x+3` so that the remainder may be 5 when divided by `(x-2)` ?

Answer

To solve the problem, we need to find the number that should be added to the polynomial \( f(x) = x^3 - 9x^2 - 2x + 3 \) so that the remainder is 5 when divided by \( (x - 2) \).

### Step-by-Step Solution:

1. **Identify the Polynomial**:
We have the polynomial \( f(x) = x^3 - 9x^2 - 2x + 3 \).

2. **Let the Number to be Added be \( c \)**:
We will denote the number we need to add as \( c \). Thus, our new polynomial becomes:
\[ f(x) + c = x^3 - 9x^2 - 2x + 3 + c \]

3. **Use the Remainder Theorem**:
According to the Remainder Theorem, the remainder of \( f(x) \) when divided by \( (x - 2) \) can be found by evaluating \( f(2) \). We want this remainder to equal 5 after adding \( c \).

4. **Calculate \( f(2) \)**:
Substitute \( x = 2 \) into \( f(x) \):
\[ f(2) = 2^3 - 9(2^2) - 2(2) + 3 \]
Calculate each term:
- \( 2^3 = 8 \)
- \( 9(2^2) = 9 \times 4 = 36 \)
- \( 2(2) = 4 \)
- Therefore, \( f(2) = 8 - 36 - 4 + 3 \)

5. **Simplify \( f(2) \)**:
Now, simplify the expression:
\[ f(2) = 8 - 36 - 4 + 3 = 8 - 36 = -28; \quad -28 - 4 = -32; \quad -32 + 3 = -29 \]
So, \( f(2) = -29 \).

6. **Set Up the Equation**:
We want:
\[ f(2) + c = 5 \]
Substitute \( f(2) \):
\[ -29 + c = 5 \]

7. **Solve for \( c \)**:
Rearranging gives:
\[ c = 5 + 29 = 34 \]

8. **Final Answer**:
Therefore, the number that should be added is:
\[ \boxed{34} \]