Find the value of a if the division of `ax^(3)+9x^(2)+4x-10` by `(x+3)` leaves a remainder 5.

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \( a \) such that the division of \( ax^3 + 9x^2 + 4x - 10 \) by \( (x + 3) \) leaves a remainder of 5, we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \( f(x) \) by \( (x - c) \) is \( f(c) \).

Here are the steps to solve the problem:

### Step 1: Identify the polynomial and the divisor
The polynomial is:
\[ f(x) = ax^3 + 9x^2 + 4x - 10 \]
The divisor is:
\[ x + 3 \]
We can rewrite the divisor as \( x - (-3) \). Therefore, we will evaluate \( f(-3) \).

### Step 2: Substitute \( x = -3 \) into the polynomial
We substitute \( x = -3 \) into \( f(x) \):
\[ f(-3) = a(-3)^3 + 9(-3)^2 + 4(-3) - 10 \]

### Step 3: Simplify the expression
Calculating each term:
- \( (-3)^3 = -27 \) so \( a(-27) = -27a \)
- \( (-3)^2 = 9 \) so \( 9(9) = 81 \)
- \( 4(-3) = -12 \)

Now substituting these values back into the equation:
\[ f(-3) = -27a + 81 - 12 - 10 \]
Combine the constant terms:
\[ f(-3) = -27a + 81 - 22 = -27a + 59 \]

### Step 4: Set the equation equal to the remainder
According to the problem, the remainder when dividing by \( (x + 3) \) is 5:
\[ -27a + 59 = 5 \]

### Step 5: Solve for \( a \)
Now, isolate \( a \):
\[ -27a + 59 = 5 \]
Subtract 59 from both sides:
\[ -27a = 5 - 59 \]
\[ -27a = -54 \]
Now, divide both sides by -27:
\[ a = \frac{-54}{-27} = 2 \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{2} \]