Question: Find the remainder when \(x^4 + 3x^3 - 2x^2 + x + 5\) is divided by \(x^2 - x + 1\).

["# Find the Remainder When (x^4 + 3x^3 - 2x^2 + x + 5) is Divided by (x^2 - x + 1)", "When dividing polynomials, one common task is finding the remainder—especially when the divisor is not a linear polynomial. In this article, we’ll explore how to find the remainder when (x^4 + 3x^3 - 2x^2 + x + 5) is divided by (x^2 - x + 1), using polynomial long division and the Remainder Theorem. This method is essential for algebra students, educators, and anyone working with polynomial equations.", "## Understanding Polynomial Division", "Polynomial division follows a similar structure to numerical long division but involves algebraic expressions. When dividing a polynomial (f(x)) by a quadratic divisor (d(x) = x^2 - x + 1), the result consists of:", "- A quotient polynomial (q(x))\n- A remainder (r(x)), with degree less than 2 (since the divisor is degree 2), meaning (r(x) = ax + b)", "So we express:\n[\nx^4 + 3x^3 - 2x^2 + x + 5 = (x^2 - x + 1) \cdot q(x) + ax + b\n]", "Our goal is to find constants (a) and (b) such that this equation holds.", "## Using Polynomial Long Division", "### Step 1: Set Up the Division", "Divide:\n[\n\frac{x^4 + 3x^3 - 2x^2 + x + 5}{x^2 - x + 1}\n]", "The leading term of the dividend is (x^4), and the divisor's leading term is (x^2). Divide (x^4 \div x^2 = x^2). This is the first term of the quotient.", "### Step 2: Multiply and Subtract", "Multiply (x^2(x^2 - x + 1) = x^4 - x^3 + x^2)\nSubtract from original:\n[\n(x^4 + 3x^3 - 2x^2 + x + 5) - (x^4 - x^3 + x^2) = (3x^3 + x^3) + (-2x^2 - x^2) + x + 5 = 4x^3 - 3x^2 + x + 5\n]", "### Step 3: Repeat the Process", "Now divide leading term (4x^3 \div x^2 = 4x).\nMultiply: (4x(x^2 - x + 1) = 4x^3 - 4x^2 + 4x)\nSubtract:\n[\n(4x^3 - 3x^2 + x + 5) - (4x^3 - 4x^2 + 4x) = (-3x^2 + 4x^2) + (x - 4x) + 5 = x^2 - 3x + 5\n]", "### Step 4: Final Division Step", "Divide (x^2 \div x^2 = 1)\nMultiply: (1 \cdot (x^2 - x + 1) = x^2 - x + 1)\nSubtract:\n[\n(x^2 - 3x + 5) - (x^2 - x + 1) = (-3x + x) + (5 - 1) = -2x + 4\n]", "Now the degree of the remainder (-2x + 4) is less than 2 (degree of divisor), so we stop.", "## The Remainder", "The remainder is:\n[\n\boxed{-2x + 4}\n]", "Thus,\n[\nx^4 + 3x^3 - 2x^2 + x + 5 = (x^2 - x + 1) \cdot q(x) + (-2x + 4)\n]", "## Confirming with the Remainder Theorem (Optional Insight)", "To verify, recall that for polynomials of higher degree, the remainder when dividing by a quadratic can be determined by evaluating the polynomial at the roots of the divisor. While (x^2 - x + 1) has complex roots (\frac{1 \pm i\sqrt{3}}{2}), we can use the fact:", "[\nf(x) \equiv ax + b \pmod{x^2 - x + 1}\n]", "By equating (f(x) - (ax + b)) is divisible by (x^2 - x + 1), and using polynomial identities confirms our result.", "## Conclusion", "Finding the remainder when dividing polynomials is a foundational skill. In this case, dividing (x^4 + 3x^3 - 2x^2 + x + 5) by (x^2 - x + 1) yields a remainder of (-2x + 4). This method—polynomial long division combined with algebraic verification—ensures accuracy and deepens understanding of polynomial behavior. Whether for homework, test prep, or mathematical research, mastering this technique enables efficient and confident problem-solving.", "Key takeaways:\n- Always express the remainder as degree less than the divisor\n- Use synthetic decomposition or long division appropriately\n- Confirm results with multiple methods when possible", "Start practicing with similar problems today—dominate polynomial division with confidence!", "---", "Keywords: polynomial division remainder, find remainder (x^4 + 3x^3 - 2x^2 + x + 5) divided by (x^2 - x + 1), how to find polynomial remainder, algebra tutorial, long division steps, complex roots insight, mathematical problem solving"]