Given that the expression `2x^3+3p x^2-4x+p` has a remainder of 5 when divided by `x+2` , find the value of `pdot`

Explanation

To solve the problem, we need to find the value of \( p \) such that the expression \( 2x^3 + 3px^2 - 4x + p \) has a remainder of 5 when divided by \( x + 2 \).

### Step-by-Step Solution:

1. **Define the Polynomial**:
Let \( f(x) = 2x^3 + 3px^2 - 4x + p \).

2. **Apply the Remainder Theorem**:
According to the Remainder Theorem, when a polynomial \( f(x) \) is divided by \( x - a \), the remainder is \( f(a) \). In this case, we are dividing by \( x + 2 \), which can be rewritten as \( x - (-2) \). Thus, we need to evaluate \( f(-2) \).

3. **Calculate \( f(-2) \)**:
\[ f(-2) = 2(-2)^3 + 3p(-2)^2 - 4(-2) + p \]
\[ = 2(-8) + 3p(4) + 8 + p \]
\[ = -16 + 12p + 8 + p \]
\[ = -16 + 8 + 13p \]
\[ = -8 + 13p \]

4. **Set the Remainder Equal to 5**:
We know from the problem statement that the remainder is 5. Therefore, we set up the equation:
\[ -8 + 13p = 5 \]

5. **Solve for \( p \)**:
\[ 13p = 5 + 8 \]
\[ 13p = 13 \]
\[ p = \frac{13}{13} = 1 \]

6. **Conclusion**:
The value of \( p \) is \( 1 \).