The polynomials `ax^(3)+3x^(2)-13` and `2x^(3)-5x+a` are divided by `(x+2)` . If the remainder in each case is same , then find the value of a .

Answer

To find the value of \( a \) such that the remainders of the polynomials \( ax^3 + 3x^2 - 13 \) and \( 2x^3 - 5x + a \) when divided by \( x + 2 \) are the same, we can use the Remainder Theorem. According to this theorem, the remainder of a polynomial \( f(x) \) when divided by \( x - c \) is \( f(c) \).

### Step-by-Step Solution:

1. **Identify the value of \( x \)**:
Since we are dividing by \( x + 2 \), we set \( x + 2 = 0 \) which gives us \( x = -2 \).

2. **Evaluate the first polynomial**:
We will evaluate \( f(x) = ax^3 + 3x^2 - 13 \) at \( x = -2 \):
\[ f(-2) = a(-2)^3 + 3(-2)^2 - 13 \]
\[ = a(-8) + 3(4) - 13 \]
\[ = -8a + 12 - 13 \]
\[ = -8a - 1 \]

3. **Evaluate the second polynomial**:
Now we evaluate \( p(x) = 2x^3 - 5x + a \) at \( x = -2 \):
\[ p(-2) = 2(-2)^3 - 5(-2) + a \]
\[ = 2(-8) + 10 + a \]
\[ = -16 + 10 + a \]
\[ = -6 + a \]

4. **Set the remainders equal**:
Since the remainders are the same, we set \( f(-2) = p(-2) \):
\[ -8a - 1 = -6 + a \]

5. **Solve for \( a \)**:
Rearranging the equation:
\[ -8a - a = -6 + 1 \]
\[ -9a = -5 \]
\[ a = \frac{5}{9} \]

### Final Answer:
The value of \( a \) is \( \frac{5}{9} \).