If the polynomials `az^3+ 4z^2+ 3z-4 and z^3-4z+ a` leave the same remainder when divided by z-3, the value of a is

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the problem, we need to find the value of \( a \) such that the two polynomials \( P_1(z) = az^3 + 4z^2 + 3z - 4 \) and \( P_2(z) = z^3 - 4z + a \) leave the same remainder when divided by \( z - 3 \).

### Step-by-Step Solution:

1. **Identify the Polynomials**:
- Let \( P_1(z) = az^3 + 4z^2 + 3z - 4 \)
- Let \( P_2(z) = z^3 - 4z + a \)

2. **Use the Remainder Theorem**:
- According to the Remainder Theorem, the remainder of a polynomial \( P(z) \) when divided by \( z - c \) is \( P(c) \).
- We will find \( P_1(3) \) and \( P_2(3) \).

3. **Calculate \( P_1(3) \)**:
\[ P_1(3) = a(3^3) + 4(3^2) + 3(3) - 4 \]
\[ = a(27) + 4(9) + 9 - 4 \]
\[ = 27a + 36 + 9 - 4 \]
\[ = 27a + 41 \]

4. **Calculate \( P_2(3) \)**:
\[ P_2(3) = (3^3) - 4(3) + a \]
\[ = 27 - 12 + a \]
\[ = 15 + a \]

5. **Set the Remainders Equal**:
Since both polynomials leave the same remainder when divided by \( z - 3 \), we set the two expressions equal:
\[ 27a + 41 = 15 + a \]

6. **Solve for \( a \)**:
- Rearranging the equation:
\[ 27a - a = 15 - 41 \]
\[ 26a = -26 \]
\[ a = -1 \]

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