If the polynomials `px ^(3) + 3x ^(2) - 3 and 2x ^(3) - 5x + p` are divided by `(x - 4),` then remainders in both the situations are equal . Find the value of p.

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the value of \( p \) such that the remainders of the two given polynomials when divided by \( (x - 4) \) are equal.

### Step-by-step Solution:

1. **Identify the Polynomials**:
We have two polynomials:
- \( P_1(x) = px^3 + 3x^2 - 3 \)
- \( P_2(x) = 2x^3 - 5x + p \)

2. **Apply the Remainder Theorem**:
According to the Remainder Theorem, the remainder of a polynomial \( P(x) \) when divided by \( (x - a) \) is \( P(a) \). Here, we will evaluate both polynomials at \( x = 4 \).

3. **Calculate the Remainder for the First Polynomial**:
\[ R_1 = P_1(4) = p(4^3) + 3(4^2) - 3 \]
\[ = p(64) + 3(16) - 3 \]
\[ = 64p + 48 - 3 \]
\[ = 64p + 45 \]

4. **Calculate the Remainder for the Second Polynomial**:
\[ R_2 = P_2(4) = 2(4^3) - 5(4) + p \]
\[ = 2(64) - 20 + p \]
\[ = 128 - 20 + p \]
\[ = 108 + p \]

5. **Set the Remainders Equal**:
Since the remainders are equal, we set \( R_1 \) equal to \( R_2 \):
\[ 64p + 45 = 108 + p \]

6. **Solve for \( p \)**:
Rearranging the equation:
\[ 64p - p = 108 - 45 \]
\[ 63p = 63 \]
\[ p = 1 \]

### Final Answer:
The value of \( p \) is \( 1 \).