When `x^(3)+3x^(2)-mx+4` is divided by x -2, the remainder is m + 3. Find the value of m.

Answer

To solve the problem, we need to find the value of \( m \) such that when the polynomial \( f(x) = x^3 + 3x^2 - mx + 4 \) is divided by \( x - 2 \), the remainder is \( m + 3 \).

### Step-by-Step Solution:

1. **Identify the Polynomial**:
We have the polynomial \( f(x) = x^3 + 3x^2 - mx + 4 \).

2. **Apply the Remainder Theorem**:
According to the Remainder Theorem, the remainder of the polynomial \( f(x) \) when divided by \( x - a \) is \( f(a) \). Here, \( a = 2 \).

3. **Calculate \( f(2) \)**:
Substitute \( x = 2 \) into the polynomial:
\[ f(2) = (2)^3 + 3(2)^2 - m(2) + 4 \]
Simplifying this:
\[ f(2) = 8 + 3 \cdot 4 - 2m + 4 \]
\[ f(2) = 8 + 12 - 2m + 4 \]
\[ f(2) = 24 - 2m \]

4. **Set the Remainder Equal to \( m + 3 \)**:
It is given that the remainder is \( m + 3 \). Therefore, we set up the equation:
\[ 24 - 2m = m + 3 \]

5. **Solve for \( m \)**:
Rearranging the equation:
\[ 24 - 3 = m + 2m \]
\[ 21 = 3m \]
Dividing both sides by 3:
\[ m = 7 \]

### Final Answer:
The value of \( m \) is \( 7 \).

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