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To solve this long division problem, we’ll divide the polynomial \( x^4 - 3x^3 + x^2 - 4x + 1 \) by \( x - 3 \) to find both the quotient \( q(x) \) and the remainder \( r \).

### Step-by-Step Solution

1. **Divide the leading term** of the dividend by the leading term of the divisor:
   \[
   \frac{x^4}{x} = x^3
   \]
   So, the first term in the quotient is \( x^3 \).

2. **Multiply \( x^3 \) by \( x - 3 \)**:
   \[
   x^3 \cdot (x - 3) = x^4 - 3x^3
   \]
   Subtract this result from the original polynomial:
   \[
   (x^4 - 3x^3 + x^2 - 4x + 1) - (x^4 - 3x^3) = x^2 - 4x + 1
   \]

3. **Repeat the process with \( x^2 \)**:
   - Divide the leading term of the remaining polynomial by the leading term of the divisor:
     \[
     \frac{x^2}{x} = x
     \]
   - Multiply \( x \) by \( x - 3 \):
     \[
     x \cdot (x - 3) = x^2 - 3x
     \]
   - Subtract this from the remaining polynomial:
     \[
     (x^2 - 4x + 1) - (x^2 - 3x) = -x + 1
     \]

4. **Repeat with the term \( -x \)**:
   - Divide the leading term of the remaining polynomial by the leading term of the divisor:
     \[
     \frac{-x}{x} = -1
     \]
   - Multiply \( -1 \) by \( x - 3 \):
     \[
     -1 \cdot (x - 3) = -x + 3
     \]
   - Subtract this from the remaining polynomial:
     \[
     (-x + 1) - (-x + 3) = -2
     \]

5. **Conclusion**:
   The quotient \( q(x) \) is \( x^3 + x - 1 \), and the remainder \( r \) is \( -2 \).

### Final Answer
- **Quotient \( q(x) = x^3 + x - 1 \)**
- **Remainder \( r = -2 \)**

Would you like additional details on any of these steps, or do you have any questions?

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Here are five related questions that might deepen your understanding:

1. How does synthetic division compare to polynomial long division for this problem?
2. How can the Remainder Theorem help verify the remainder in polynomial division?
3. What are the steps to divide a polynomial by a quadratic divisor?
4. Can the quotient be further factored or simplified?
5. How can polynomial division be applied in real-world scenarios, like in physics or engineering?

**Tip:** Remember that the Remainder Theorem states that the remainder of dividing a polynomial \( f(x) \) by \( x - a \) is simply \( f(a) \). This can be a quick way to check the remainder in simple cases.

Perform the long division \( \frac{x^4 - 3x^3 + x^2 - 4x + 1}{x - 3} \) and identify the quotient and remainder.

Learn how to perform long division on the polynomial \( x^4 - 3x^3 + x^2 - 4x + 1 \) divided by \( x - 3 \) to identify the quotient and remainder. This solution uses polynomial division techniques and applies the Remainder Theorem. Suitable for high school math students in Grades 10-12.

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