If the polynomials `ax^(3)+4x^(2)+3x-4` and `x^(3)-4x+a` leave the same remainder when divided by `(x-3)`, find the value of a .

Answer

To find the value of \( a \) such that the polynomials \( P(x) = ax^3 + 4x^2 + 3x - 4 \) and \( Q(x) = x^3 - 4x + a \) leave the same remainder when divided by \( (x - 3) \), we will use the Remainder Theorem.

### Step-by-step Solution:

1. **Apply the Remainder Theorem**: According to the Remainder Theorem, the remainder of a polynomial \( f(x) \) when divided by \( (x - c) \) is \( f(c) \). Here, we need to evaluate both polynomials at \( x = 3 \).

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

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

4. **Set the remainders equal**:
Since both polynomials leave the same remainder when divided by \( (x - 3) \), we set \( P(3) = Q(3) \):
\[ 27a + 41 = 15 + a \]

5. **Solve for \( a \)**:
\[ 27a - a = 15 - 41 \]
\[ 26a = -26 \]
\[ a = -1 \]

Thus, the value of \( a \) is \( -1 \).