Question: Find the remainder when $ x^5 - 2x^3 + x^2 + 1 $ is divided by $ x^2 - 3x + 2 $, focusing on polynomial division techniques.

["# Find the Remainder When ( x^5 - 2x^3 + x^2 + 1 ) is Divided by ( x^2 - 3x + 2 ): Polynomial Division Techniques Explained", "Polynomial division is a fundamental concept in algebra, often appearing in both academic work and applied mathematics. One common question students encounter is: Find the remainder when ( x^5 - 2x^3 + x^2 + 1 ) is divided by ( x^2 - 3x + 2 ). Mastering this process not only helps solve division problems efficiently but also reinforces understanding of the Remainder Theorem and polynomial long division.", "This article walks through step-by-step how to divide ( x^5 - 2x^3 + x^2 + 1 ) by ( x^2 - 3x + 2 ) using standard polynomial division techniques, clearly explaining each phase so readers of all levels can follow and apply the method.", "## Understanding the Problem", "We are tasked with dividing a polynomial ( f(x) = x^5 - 2x^3 + x^2 + 1 ) by a quadratic divisor ( d(x) = x^2 - 3x + 2 ). According to polynomial division rules, when dividing a polynomial ( f(x) ) by a non-zero polynomial ( d(x) ), the result is:", "[\nf(x) = q(x) \cdot d(x) + r(x)\n]", "Where:\n- ( q(x) ) is the quotient polynomial,\n- ( r(x) ) is the remainder polynomial with degree less than that of ( d(x) ) (i.e., degree < 2).", "Because the divisor is degree 2, the remainder must be at most degree 1. Thus, we expect:", "[\nr(x) = ax + b\n]", "Our goal is to find real numbers ( a ) and ( b ) such that:", "[\nx^5 - 2x^3 + x^2 + 1 = (x^2 - 3x + 2) \cdot q(x) + ax + b\n]", "## Step 1: Factor the Divisor", "Before proceeding with polynomial long division, simplify the divisor by factoring:", "[\nx^2 - 3x + 2 = (x - 1)(x - 2)\n]", "This factorization is crucial because it lets us apply the Chinese Remainder Theorem for polynomials: we can separately find the remainder modulo ( x - 1 ) and modulo ( x - 2 ), then combine the results to determine ( ax + b ).", "## Step 2: Apply the Remainder Evaluation Trick", "Instead of performing full polynomial division, we use the fact that the remainder when dividing by a quadratic with distinct linear factors is completely determined by the values at ( x = 1 ) and ( x = 2 ).", "Let:", "[\nf(x) = x^5 - 2x^3 + x^2 + 1\n]", "Then:", "[\nf(x) \equiv r(x) \pmod{x^2 - 3x + 2}\n]", "Because ( r(x) ) is linear, suppose:", "[\nr(x) = ax + b\n]", "Using the roots of the divisor:\n- At ( x = 1 ): ( f(1) = r(1) = a(1) + b = a + b )\n- At ( x = 2 ): ( f(2) = r(2) = a(2) + b = 2a + b )", "So we compute ( f(1) ) and ( f(2) ) to form two equations.", "### Compute ( f(1) ):", "[\nf(1) = (1)^5 - 2(1)^3 + (1)^2 + 1 = 1 - 2 + 1 + 1 = 1\n]\nSo:\n[\na + b = 1 \quad \ ext{(Equation 1)}\n]", "### Compute ( f(2) ):", "[\nf(2) = (2)^5 - 2(2)^3 + (2)^2 + 1 = 32 - 16 + 4 + 1 = 21\n]\nSo:\n[\n2a + b = 21 \quad \ ext{(Equation 2)}\n]", "## Step 3: Solve the System of Equations", "From Equation 1: ( b = 1 - a )", "Substitute into Equation 2:", "[\n2a + (1 - a) = 21\n]\n[\na + 1 = 21\n]\n[\na = 20\n]", "Then ( b = 1 - 20 = -19 )", "Thus, the remainder is:", "[\nr(x) = 20x - 19\n]", "## Step 4: Verification via Polynomial Division (Optional but Recommended)", "To confirm, perform polynomial division using long division:", "Divide ( x^5 + 0x^4 - 2x^3 + x^2 + 0x + 1 ) by ( x^2 - 3x + 2 ).", "The process confirms that after full division, the remainder is indeed ( 20x - 19 ), validating our earlier method.", "## Why This Technique Works", "This method leverages:\n- Polynomial remainder theorem: The remainder depends only on values at specific points.\n- Chinese Remainder Theorem for polynomials: When divisor factors into coprime linear terms, remainder is fully determined by evaluations at roots.\n- Efficiency: Avoids lengthy long division when remainder degree is low.", "## Conclusion", "Finding the remainder when dividing ( x^5 - 2x^3 + x^2 + 1 ) by ( x^2 - 3x + 2 ) can be efficiently solved using either full polynomial division or by evaluating at the divisor’s roots. The first method—while more mechanical—is reliable; the second exploits key algebraic theorems and is faster when remainder degree is small.", "By mastering these techniques, students build a deeper understanding of polynomial algebra and prepare for advanced topics in calculus, differential equations, and control theory.", "---", "Keywords: polynomial division, remainder theorem, long division of polynomials, ( x^2 - 3x + 2 ), finding remainder, algebra techniques, polynomial remainder, divisor factorization, Chinese Remainder Theorem, math tutorial.", "For further study, explore similar problems with cubic and quartic dividends, and try dividing by higher-degree polynomials using the same root-based remainder approach."]