https://file.aiquickdraw.com/mathbot/file/17259676446617d59pd0o.jpg

I see the uploaded image contains polynomial division problems that need to be solved using the long division method. Let’s go through the first one as an example:

### Problem 1: 
\[ \frac{x^3 - 8x^2 + 17x - 10}{x - 2} \]

#### Step-by-Step Long Division:
1. **First term:**
   - Divide the leading term \( x^3 \) by \( x \). The result is \( x^2 \).
   - Multiply \( x^2 \) by \( x - 2 \), giving \( x^3 - 2x^2 \).
   - Subtract \( (x^3 - 2x^2) \) from \( (x^3 - 8x^2) \), which results in \( -6x^2 \).

2. **Second term:**
   - Divide \( -6x^2 \) by \( x \). The result is \( -6x \).
   - Multiply \( -6x \) by \( x - 2 \), giving \( -6x^2 + 12x \).
   - Subtract \( (-6x^2 + 12x) \) from \( (-6x^2 + 17x) \), which results in \( 5x \).

3. **Third term:**
   - Divide \( 5x \) by \( x \). The result is \( 5 \).
   - Multiply \( 5 \) by \( x - 2 \), giving \( 5x - 10 \).
   - Subtract \( (5x - 10) \) from \( (5x - 10) \), which results in \( 0 \).

### Final Answer:
\[
x^2 - 6x + 5
\]

This process can be repeated for the other problems in the image.

Would you like me to explain the steps for the other questions? Let me know if you have any questions about this method.

Here are five follow-up questions:
1. How do you know when to stop the division in polynomial long division?
2. What is the degree of the quotient compared to the dividend and divisor in polynomial long division?
3. How do you handle polynomials with missing terms (like \( x^4 + 0x^3 \))?
4. Can you perform synthetic division for the problems where the divisor is linear (e.g., \( x - 2 \))?
5. How does long division differ from synthetic division in terms of steps and ease of use?

**Tip:** Always align the terms by their degree when performing polynomial division to avoid confusion.

Divide the polynomial using the long division method: (x^3 - 8x^2 + 17x - 10) ÷ (x - 2)

Learn how to solve the polynomial division (x^3 - 8x^2 + 17x - 10) ÷ (x - 2) using the long division method. Step-by-step explanations are provided, covering key algebraic concepts for Grade 10 students.

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