We are to divide \( u^5 - 3u^3 + 2u + 8 \) by \( u^2 - 2u + 1 \). Note that:

["Title: Polynomial Division Explained: Dividing ( u^5 - 3u^3 + 2u + 8 ) by ( u^2 - 2u + 1 )", "---", "### Introduction", "Polynomial division is a fundamental operation in algebra, similar to numerical long division but applied to expressions involving variables. In this article, we walk through the division of ( u^5 - 3u^3 + 2u + 8 ) by ( u^2 - 2u + 1 ), explaining each step clearly. We use polynomial division with the goal of expressing the dividend as:", "[\n\frac{u^5 - 3u^3 + 2u + 8}{u^2 - 2u + 1} = Q(u) + \frac{R(u)}{u^2 - 2u + 1}\n]", "where ( Q(u) ) is the quotient and ( R(u) ) is the remainder, with the degree of ( R ) strictly less than that of the divisor.", "This guide will help students, educators, and math enthusiasts master polynomial division by breaking down the process into clear, logical steps—ideal for dominating this topic in math coursework or self-study.", "---", "### Step 1: Understand the Divisor and Divend", "We begin by identifying the dividend and divisor:", "- Dividend: ( u^5 - 3u^3 + 0u^2 + 2u + 8 )\n (Note: Add the missing ( 0u^2 ) term for completeness.)", "- Divisor: ( u^2 - 2u + 1 )\n This is a quadratic polynomial that factors as ( (u - 1)^2 ), a key observation that can simplify long division.", "---", "### Step 2: Perform Polynomial Long Division", "We divide ( u^5 - 3u^3 + 0u^2 + 2u + 8 ) by ( u^2 - 2u + 1 ) using long division.", "#### Step 2.1: Divide Leading Terms\nDivide the leading term of the dividend ( u^5 ) by the leading term of the divisor ( u^2 ):", "[\n\frac{u^5}{u^2} = u^3\n]", "This is the first term of the quotient ( Q(u) ).", "#### Step 2.2: Multiply and Subtract\nMultiply the entire divisor by ( u^3 ):", "[\nu^3(u^2 - 2u + 1) = u^5 - 2u^4 + u^3\n]", "Subtract this from the dividend:", "[\n(u^5 - 3u^3 + 0u^2 + 2u + 8) - (u^5 - 2u^4 + u^3) = 2u^4 - 4u^3 + 0u^2 + 2u + 8\n]", "#### Step 2.3: Repeat the Process\nNow divide ( 2u^4 ) by ( u^2 ):", "[\n\frac{2u^4}{u^2} = 2u^2\n]", "Multiply divisor by ( 2u^2 ):", "[\n2u^2(u^2 - 2u + 1) = 2u^4 - 4u^3 + 2u^2\n]", "Subtract:", "[\n(2u^4 - 4u^3 + 0u^2 + 2u + 8) - (2u^4 - 4u^3 + 2u^2) = -2u^2 + 2u + 8\n]", "#### Step 2.4: Continue Division\nDivide ( -2u^2 ) by ( u^2 ):", "[\n\frac{-2u^2}{u^2} = -2\n]", "Multiply divisor by (-2):", "[\n-2(u^2 - 2u + 1) = -2u^2 + 4u - 2\n]", "Subtract:", "[\n(-2u^2 + 2u + 8) - (-2u^2 + 4u - 2) = (2u - 4u) + (8 + 2) = -2u + 10\n]", "---", "### Step 3: Final Result", "At this point, the degree of the remainder (-2u + 10) (degree 1) is less than that of the divisor (degree 2), so we stop.", "Thus, we write:", "[\n\frac{u^5 - 3u^3 + 2u + 8}{u^2 - 2u + 1} = u^3 + 2u^2 - 2 + \frac{-2u + 10}{u^2 - 2u + 1}\n]", "---", "### Step 4: Summary and Remainder", "- Quotient: ( Q(u) = u^3 + 2u^2 - 2 )\n- Remainder: ( R(u) = -2u + 10 )\n- Division expression:", "[\nu^5 - 3u^3 + 2u + 8 = (u^2 - 2u + 1)(u^3 + 2u^2 - 2) + (-2u + 10)\n]", "---", "### Why This Matters", "Understanding polynomial division is essential in algebra, calculus, and engineering applications. It underpins partial fraction decomposition, solving differential equations, and algorithmic polynomial processing.", "---", "### Key Takeaways", "- Always write missing degree terms (e.g., ( 0u^2 )) for complete expression.\n- Long division mirrors numerical long division: divide, multiply, subtract, repeat.\n- The remainder’s degree must be less than the divisor’s.\n- The quotient and remainder uniquely decompose the dividend with respect to the divisor.", "---", "### Final Thoughts", "Dividing ( u^5 - 3u^3 + 2u + 8 ) by ( u^2 - 2u + 1 ) may seem complex at first, but with careful, step-by-step division, the process becomes manageable. Practice with real polynomials helps reinforce these skills—whether for exams, homework, or deeper mathematical insight.", "Mastering polynomial division opens doors to advanced math and applications. Keep practicing, and this powerful technique will become second nature!", "---", "Keywords: Polynomial division, divide ( u^5 - 3u^3 + 2u + 8 ) by ( u^2 - 2u + 1 ), long division of polynomials, remainder and quotient, algebra tutorial, divide polynomials step-by-step."]