Misal f(x) = ax³ + 4x² - 8x + a + 9 habis dibagi x – 2 dan hasil baginya adalah h(x). jika h(x) dibagi (x + 1) maka sisa pembaginya adalah …

A.      – 18

B.      – 7

C.      – 1

D.      1

E.      7

To solve this problem, we need to apply the concept of the remainder theorem and synthetic division.

### Step-by-step Solution:

1. **Given Polynomial:**
   \( f(x) = ax^3 + 4x^2 - 8x + a + 9 \)

2. **Condition 1: \( f(x) \) is divisible by \( x - 2 \):**

   If \( f(x) \) is divisible by \( x - 2 \), then \( x = 2 \) is a root of \( f(x) \). Thus, \( f(2) = 0 \).

   Let's find \( f(2) \):

   \[
   f(2) = a(2)^3 + 4(2)^2 - 8(2) + a + 9
   \]
   \[
   f(2) = 8a + 16 - 16 + a + 9
   \]
   \[
   f(2) = 9a + 9
   \]

   Since \( f(2) = 0 \), we have:
   
   \[
   9a + 9 = 0
   \]
   \[
   9a = -9
   \]
   \[
   a = -1
   \]

3. **Substitute \( a = -1 \) into \( f(x) \):**

   Now, replace \( a \) in \( f(x) \):

   \[
   f(x) = -x^3 + 4x^2 - 8x - 1 + 9
   \]
   \[
   f(x) = -x^3 + 4x^2 - 8x + 8
   \]

4. **Divide \( f(x) \) by \( x - 2 \) using synthetic division to find \( h(x) \):**

   We know that the remainder when \( f(x) \) is divided by \( x - 2 \) is zero because \( x = 2 \) is a root. Let's perform synthetic division:

   - Coefficients of \( f(x) \): \(-1, 4, -8, 8\)
   - Divide by \( x - 2 \) (\( x = 2 \))

   Synthetic division steps:
   - Bring down \(-1\).
   - Multiply by \(2\), add to next coefficient:
     \[
     -1, \quad (-1 \times 2 + 4) = 2, \quad (2 \times 2 - 8) = -4, \quad (-4 \times 2 + 8) = 0
     \]

   So, the quotient polynomial \( h(x) \) is:
   \[
   h(x) = -x^2 + 2x - 4
   \]

5. **Condition 2: Find the remainder when \( h(x) \) is divided by \( x + 1 \):**

   To find the remainder when \( h(x) \) is divided by \( x + 1 \), we use the remainder theorem again. Substitute \( x = -1 \) into \( h(x) \):

   \[
   h(-1) = -(-1)^2 + 2(-1) - 4
   \]
   \[
   h(-1) = -1 - 2 - 4
   \]
   \[
   h(-1) = -7
   \]

6. **Conclusion:**

   The remainder when \( h(x) \) is divided by \( x + 1 \) is \(-7\).

   **Answer: B. -7**

Would you like more details or have any questions? Here are some related topics you might find interesting:

1. How to perform synthetic division?
2. What is the remainder theorem?
3. How do you find roots of a polynomial?
4. What are the different methods for polynomial division?
5. How does changing the coefficient affect the roots of a polynomial?

**Tip:** The Remainder Theorem is a powerful tool for evaluating polynomials quickly without fully expanding them, especially useful for checking divisibility and finding remainders.

Explore how to solve a polynomial division problem using the remainder theorem and synthetic division. This problem involves finding the remainder when dividing h(x) by (x + 1) after determining h(x) from a given polynomial f(x).

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