The polynomial
`f (x) = ax ^(3) + 9x ^(2) + 4x - 8,` when divided by (x+3) leaves the remainder (-20). Then , the value of a is

Options

A. 1
B. 2
C. 3
D. 5

Answer

To find the value of \( a \) in the polynomial \( f(x) = ax^3 + 9x^2 + 4x - 8 \) when it is divided by \( (x + 3) \) and leaves a remainder of \( -20 \), we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial \( f(x) \) by \( (x - c) \) is equal to \( f(c) \).

### Step-by-step solution:

1. **Identify the polynomial and the divisor**:
The polynomial is given as:
\[ f(x) = ax^3 + 9x^2 + 4x - 8 \]
The divisor is \( (x + 3) \), which means we will evaluate \( f(-3) \).

2. **Set up the equation using the Remainder Theorem**:
According to the problem, when \( f(x) \) is divided by \( (x + 3) \), the remainder is \( -20 \). Therefore, we can write:
\[ f(-3) = -20 \]

3. **Substitute \( x = -3 \) into the polynomial**:
Now, let's substitute \( -3 \) into the polynomial:
\[ f(-3) = a(-3)^3 + 9(-3)^2 + 4(-3) - 8 \]
Simplifying this:
\[ f(-3) = a(-27) + 9(9) - 12 - 8 \]
\[ = -27a + 81 - 12 - 8 \]
\[ = -27a + 81 - 20 \]
\[ = -27a + 61 \]

4. **Set the equation equal to -20**:
Now, we set the expression equal to \( -20 \):
\[ -27a + 61 = -20 \]

5. **Solve for \( a \)**:
Rearranging the equation:
\[ -27a = -20 - 61 \]
\[ -27a = -81 \]
Dividing both sides by \( -27 \):
\[ a = \frac{-81}{-27} = 3 \]

6. **Conclusion**:
Therefore, the value of \( a \) is:
\[ a = 3 \]