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The result of dividing \( P(x) = 3x^5 + x^3 - 3x^2 + 4x - 8 \) by \( D(x) = x^2 - 4x + 1 \) using polynomial long division is:

\[
\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}
\]

Where:

- \( Q(x) = 3x^3 + 12x^2 + 46x + 169 \)
- \( R(x) = 634x - 177 \)

Thus, the expression becomes:

\[
\frac{P(x)}{D(x)} = 3x^3 + 12x^2 + 46x + 169 + \frac{634x - 177}{x^2 - 4x + 1}
\]

Would you like a detailed explanation of the division process, or do you have any further questions?

Here are 5 related questions you may find helpful:
1. How do you perform synthetic division for polynomials?
2. What is the significance of the quotient and remainder in polynomial division?
3. Can you explain how polynomial long division is similar to numerical long division?
4. How do you check if a polynomial division is correct?
5. What happens when the remainder is zero in polynomial division?

**Tip:** Always simplify the remainder over the divisor if possible to make the final expression cleaner.

Two polynomials P(x) = 3x^5 + x^3 - 3x^2 + 4x - 8 and D(x) = x^2 - 4x + 1 are given. Use synthetic or long division to divide P(x) by D(x), and express the quotient in the form P(x)/D(x) = Q(x) + R(x)/D(x).

Learn how to divide the polynomials P(x) = 3x^5 + x^3 - 3x^2 + 4x - 8 by D(x) = x^2 - 4x + 1 using long division or synthetic division. This step-by-step solution includes the quotient and remainder in the form P(x)/D(x) = Q(x) + R(x)/D(x). Suitable for Grades 10-12.

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