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To solve this polynomial division problem, we'll use polynomial long division. Let's go through it step-by-step for:

\[
3x^3 - x^2 + 5 \quad \text{divided by} \quad x^2 - 4x + 1
\]

### Step 1: Set up the division
We want to divide the dividend \( 3x^3 - x^2 + 0x + 5 \) by the divisor \( x^2 - 4x + 1 \).

### Step 2: Divide the leading terms
The leading term of the dividend is \( 3x^3 \), and the leading term of the divisor is \( x^2 \). Divide \( 3x^3 \) by \( x^2 \) to get \( 3x \).

### Step 3: Multiply and subtract
Multiply \( 3x \) by \( x^2 - 4x + 1 \):
\[
3x \cdot (x^2 - 4x + 1) = 3x^3 - 12x^2 + 3x
\]
Now, subtract this from the original polynomial:
\[
(3x^3 - x^2 + 0x + 5) - (3x^3 - 12x^2 + 3x) = 11x^2 - 3x + 5
\]

### Step 4: Repeat the process
The new leading term is \( 11x^2 \). Divide \( 11x^2 \) by \( x^2 \) to get \( 11 \).

### Step 5: Multiply and subtract again
Multiply \( 11 \) by \( x^2 - 4x + 1 \):
\[
11 \cdot (x^2 - 4x + 1) = 11x^2 - 44x + 11
\]
Subtract this from the remainder:
\[
(11x^2 - 3x + 5) - (11x^2 - 44x + 11) = 41x - 6
\]

### Final Answer
So, the quotient is \( 3x + 11 \) and the remainder is \( 41x - 6 \).

Thus, we can express this as:
\[
3x^3 - x^2 + 5 = (x^2 - 4x + 1)(3x + 11) + (41x - 6)
\]

Would you like further details on any step, or have any questions?

### Further Questions
1. What is polynomial long division, and how is it similar to regular long division?
2. How can we verify the quotient and remainder are correct?
3. Are there alternative methods for dividing polynomials, like synthetic division?
4. How does the remainder theorem apply to this problem?
5. What happens if the divisor is a higher degree polynomial than the dividend?

**Tip:** When dividing polynomials, always align terms by descending powers of \( x \) to avoid mistakes during subtraction.

Find the quotient and remainder of (3x^3 - x^2 + 5) ÷ (x^2 - 4x + 1). Enter the correct value in the box. Give your answer in the form dividend = (divisor)(quotient) + remainder.

Learn how to find the quotient and remainder of (3x^3 - x^2 + 5) divided by (x^2 - 4x + 1) using polynomial long division. This step-by-step solution explains key algebraic concepts, formulas, and theorems relevant to polynomial division. Suitable for Grades 10-12.

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