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### Question 9: Synthetic Division
We need to divide \( x^3 + 3x^2 + 9x + 17 \) by \( x + 2 \) using synthetic division.

1. **Set up synthetic division:**
   The divisor \( x + 2 \) has a root \( x = -2 \), so we'll use -2 in synthetic division. The coefficients of the polynomial are \( 1, 3, 9, 17 \).

   Here's how synthetic division proceeds:
   \[
   \begin{array}{r|rrrr}
   -2 & 1 & 3 & 9 & 17 \\
     &   & -2 & -2 & -14 \\
   \hline
     & 1 & 1 & 7 & 3 \\
   \end{array}
   \]

   So, the quotient is \( x^2 + x + 7 \) and the remainder is 3.

   **Answer:**
   - Quotient: \( x^2 + x + 7 \)
   - Remainder: 3

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### Question 10: Long Division
We need to divide \( x^3 - 7x^2 + 17 \) by \( x - 2 \) using long division.

1. Divide \( x^3 \) by \( x \), which gives \( x^2 \).
2. Multiply \( x^2 \) by \( x - 2 \) to get \( x^3 - 2x^2 \), and subtract this from the original polynomial to get \( -5x^2 + 17 \).
3. Divide \( -5x^2 \) by \( x \), which gives \( -5x \).
4. Multiply \( -5x \) by \( x - 2 \) to get \( -5x^2 + 10x \), and subtract this from the current polynomial to get \( -10x + 17 \).
5. Divide \( -10x \) by \( x \), which gives \( -10 \).
6. Multiply \( -10 \) by \( x - 2 \) to get \( -10x + 20 \), and subtract this from the current polynomial to get a remainder of \( -3 \).

   **Answer:**
   - Quotient: \( x^2 - 5x - 10 \)
   - Remainder: \( -3 \)

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Here are 5 related questions to expand on these concepts:

1. How does synthetic division work for higher-degree polynomials?
2. Can synthetic division be used for divisors of forms other than \( x + c \)?
3. How do you check the result of synthetic division by multiplying back the quotient and divisor?
4. What is the relationship between the remainder in division and the remainder theorem?
5. How is synthetic division different from polynomial long division?

**Tip:** Synthetic division is much faster for divisors in the form \( x + c \), but for more complex divisors, long division is a more reliable method.

Find the quotient and remainder using synthetic division for (x^3 + 3x^2 + 9x + 17) / (x + 2) and long division for (x^3 - 7x^2 + 17) / (x - 2).

This page provides detailed solutions for finding the quotient and remainder of polynomial divisions using synthetic and long division methods. Perfect for high school students learning Algebra and Polynomial Division techniques.

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