Which number should be added to `2x^(3)-3x^(2)+x` so that when the resulting polynomial is divided by x-2, the remainder is 3?

Answer

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

### Step-by-step Solution:

1. **Define the Polynomial**:
Let \( f(x) = 2x^3 - 3x^2 + x \).

2. **Let the Number to be Added be \( c \)**:
We want to find the value of \( c \) such that when we add it to \( f(x) \), the new polynomial \( f(x) + c \) gives a remainder of 3 when divided by \( x - 2 \).

3. **Use the Remainder Theorem**:
According to the Remainder Theorem, the remainder of a polynomial \( f(x) \) when divided by \( x - a \) is \( f(a) \). Here, we need to find \( f(2) \).

4. **Calculate \( f(2) \)**:
\[ f(2) = 2(2^3) - 3(2^2) + 2 \]
\[ = 2(8) - 3(4) + 2 \]
\[ = 16 - 12 + 2 \]
\[ = 6 \]

5. **Set Up the Equation**:
We want the remainder when \( f(x) + c \) is divided by \( x - 2 \) to be 3. Thus, we set up the equation:
\[ f(2) + c = 3 \]
Substituting \( f(2) \):
\[ 6 + c = 3 \]

6. **Solve for \( c \)**:
\[ c = 3 - 6 \]
\[ c = -3 \]

### Conclusion:
The number that should be added to the polynomial \( 2x^3 - 3x^2 + x \) so that the remainder when divided by \( x - 2 \) is 3 is \( -3 \).