Question: Find the remainder when $ x^5 - 3x^3 + 2x - 1 $ is divided by $ x^2 - 2x + 1 $.

["SEO-Optimized Article: How to Find the Remainder When $ x^5 - 3x^3 + 2x - 1 $ Is Divided by $ x^2 - 2x + 1 $", "Title:\nHow to Find the Remainder of $ x^5 - 3x^3 + 2x - 1 $ Divided by $ x^2 - 2x + 1 $ – Step-by-Step Guide with Remainder Explanation", "Meta Description:\nLearn how to find the remainder when dividing $ x^5 - 3x^3 + 2x - 1 $ by $ x^2 - 2x + 1 $ using polynomial long division and the Remainder Theorem. A clear, step-by-step guide for algebraic problems.", "---", "### Introduction", "Dividing polynomials using division algorithms is a fundamental skill in algebra. A common question in math exams and competitions is:\nFind the remainder when $ x^5 - 3x^3 + 2x - 1 $ is divided by $ x^2 - 2x + 1 $.", "In this article, we’ll walk through the process using polynomial long division and key theorems, including the Remainder Theorem and Factor Theorem, to simplify the problem. Whether you're a student, teacher, or math enthusiast, this step-by-step guide will help you master polynomial division and remainder finding.", "---", "### Understanding the Divisor: $ x^2 - 2x + 1 $", "Before diving into division, note that the divisor can be factored:", "$$\nx^2 - 2x + 1 = (x - 1)^2\n$$", "Since the divisor is a quadratic, the remainder $ R(x) $ will be of degree less than 2 — that is, it will be a linear polynomial of the form:", "$$\nR(x) = ax + b\n$$", "Our goal: Find constants $ a $ and $ b $ such that:", "$$\nx^5 - 3x^3 + 2x - 1 = (x - 1)^2 \cdot Q(x) + ax + b\n$$", "for some quadratic polynomial $ Q(x) $.", "---", "### Method 1: Polynomial Long Division", "Polynomial long division works similarly to numerical long division, but with variables.", "Step 1: Set up the division", "Divide $ x^5 + 0x^4 - 3x^3 + 0x^2 + 2x - 1 $ by $ x^2 - 2x + 1 $", "Step 2: Divide leading terms\nEstimate how many times $ x^2 $ goes into $ x^5 $: $ x^3 $", "Multiply:\n$$\nx^3(x^2 - 2x + 1) = x^5 - 2x^4 + x^3\n$$", "Subtract:\n$$\n(x^5 + 0x^4 - 3x^3 + \cdots) - (x^5 - 2x^4 + x^3) = 2x^4 - 4x^3 + \cdots\n$$", "Step 3: Repeat", "Now divide $ 2x^4 $ by $ x^2 $: get $ 2x^2 $", "Multiply:\n$$\n2x^2(x^2 - 2x + 1) = 2x^4 - 4x^3 + 2x^2\n$$", "Subtract:\n$$\n(2x^4 - 4x^3 + 0x^2) - (2x^4 - 4x^3 + 2x^2) = -2x^2 + 0x\n$$", "Bring down next terms: $ -2x^2 + 2x - 1 $", "Now divide $ -2x^2 $ by $ x^2 $: get $ -2 $", "Multiply:\n$$\n-2(x^2 - 2x + 1) = -2x^2 + 4x - 2\n$$", "Subtract:\n$$\n(-2x^2 + 2x - 1) - (-2x^2 + 4x - 2) = -2x + 1\n$$", "We now have:", "- Quotient: $ x^3 + 2x^2 - 2 $\n- Remainder: $ -2x + 1 $", "✅ Since the degree of the remainder $ (-2x + 1) $ is less than 2, the division stops here.", "---", "### Verification Using the Remainder Theorem Approach", "To confirm, we can use the fact that remainder when dividing by $ (x - 1)^2 $ is $ ax + b $, and satisfy the conditions:", "- $ P(1) = a(1) + b $\n- $ P'(1) = a $ (by differentiable remainder form at repeated root)", "Let $ P(x) = x^5 - 3x^3 + 2x - 1 $", "Step 1: Compute $ P(1) $:", "$$\nP(1) = (1)^5 - 3(1)^3 + 2(1) - 1 = 1 - 3 + 2 - 1 = -1\n$$\nSo:\n$$\na + b = -1 \quad \ ext{(Equation 1)}\n$$", "Step 2: Compute $ P'(x) $:", "$$\nP'(x) = 5x^4 - 9x^2 + 2\n$$", "Evaluate at $ x = 1 $:", "$$\nP'(1) = 5(1)^4 - 9(1)^2 + 2 = 5 - 9 + 2 = -2\n$$", "But since divisor is $ (x - 1)^2 $, the derivative of the remainder $ R(x) = ax + b $ must match $ P'(1) $ up to the multiplier from the divisor. Alternatively, using the known method:\n$$\nR'(x) = a \Rightarrow R'(1) = a\n$$\nAnd since $ P(x) = (x - 1)^2 Q(x) + ax + b $, then:", "$$\nP'(x) = 2(x - 1)Q(x) + (x - 1)^2 Q'(x) + a\n\Rightarrow P'(1) = a\n$$", "So:\n$$\na = P'(1) = -2\n$$", "Substitute into Equation 1:\n$$\n-2 + b = -1 \Rightarrow b = 1\n$$", "Thus, remainder is:", "$$\nR(x) = -2x + 1\n$$", "This matches the result from polynomial division — confirming correctness.", "---", "### Final Answer", "The remainder when $ x^5 - 3x^3 + 2x - 1 $ is divided by $ x^2 - 2x + 1 $ is:", "$$\n\boxed{-2x + 1}\n$$", "---", "### Conclusion", "Finding the remainder of polynomial division can be efficiently done using long division or the Remainder Theorem tailored for repeated roots. By recognizing that the remainder is linear and applying both algebraic verification and derivative insight, we confirmed the result systematically.", "Pro Tip: For any divisor of degree 2, always expect a linear remainder. Use either long division or remnant evaluation at the root and its derivative to double-check your work.", "---", "### Frequently Asked Questions (FAQ)", "Q: Why is the remainder linear?\nA: Because the divisor is degree 2; the remainder must be of degree less than 2, i.e., linear or constant.", "Q: Can I use synthetic division for $ (x - 1)^2 $?\nA: Synthetic division is limited to linear divisors. For higher-degree divisors, standard polynomial division or remainder theorem extensions are needed.", "Q: How does the multiplicity of the root affect the remainder?\nA: At each repeated root, the derivative of the remainder must match the derivative of $ P(x) $ at that point — hence why $ R'(1) = P'(1) $.", "---", "### Keywords for SEO Optimization:\npolynomial division remainder, divide $ x^5 - 3x^3 + 2x - 1 by $ $ x^2 - 2x + 1 $, remainder when dividing by $ (x-1)^2 $, remainder theorem explained, how to find polynomial remainder, step-by-step polynomial division, algebraic remainder verification", "---", "Don’t forget to practice with your own problems — mastering remainders improves fluency in algebra and calculus readiness!"]