Find the remainder when \( u^5 - 3u^3 + 2u + 8 \) is divided by \( u^2 - 2u + 1 \).

["# Finding the Remainder When ( u^5 - 3u^3 + 2u + 8 ) is Divided by ( u^2 - 2u + 1 )", "When dividing polynomials, it’s essential to understand both the quotient and especially the remainder, which simplifies division in algebraic expressions and is key in polynomial division, system modeling, and error checking. This article breaks down how to efficiently find the remainder when dividing:", "[\nu^5 - 3u^3 + 2u + 8 \quad \ ext{by} \quad u^2 - 2u + 1\n]", "---", "## Understanding Polynomial Division and Remainders", "Polynomial division follows rules similar to numerical long division. When dividing a polynomial ( f(u) ) by a quadratic divisor ( d(u) = u^2 - 2u + 1 ), the remainder ( R(u) ) must be of lower degree than the divisor. Since ( u^2 - 2u + 1 ) is degree 2, the remainder will be at most linear:", "[\nR(u) = au + b\n]", "This means the result of the division looks like:", "[\nf(u) = (u^2 - 2u + 1) \cdot Q(u) + (au + b)\n]", "where ( Q(u) ) is the quotient polynomial.", "---", "## Step 1: Factor the Divisor", "First, notice that:", "[\nu^2 - 2u + 1 = (u - 1)^2\n]", "This helps simplify calculations, especially when applying polynomial remainder theorems or substitution.", "---", "## Step 2: Use the Remainder Theorem for Higher Degrees", "Because direct long division of ( u^5 - 3u^3 + 2u + 8 ) by ( (u - 1)^2 ) can be lengthy, we use a powerful method involving differentiation and substitution—deriving conditions for both the remainder and its derivative at the root ( u = 1 ).", "Let:\n( f(u) = u^5 - 3u^3 + 2u + 8 )\n( d(u) = (u - 1)^2 )\n( R(u) = au + b )", "Then, dividing by ( (u - 1)^2 ) gives remainder ( R(u) ), so we have:", "[\nf(u) = (u - 1)^2 Q(u) + R(u)\n]", "At ( u = 1 ), both sides give:", "[\nf(1) = R(1) \quad \ ext{and} \quad f'(1) = R'(1)\n]", "This gives a system of two equations to solve for ( a ) and ( b ).", "---", "## Step 3: Compute ( f(1) )", "Evaluate ( f(1) ):", "[\nf(1) = (1)^5 - 3(1)^3 + 2(1) + 8 = 1 - 3 + 2 + 8 = 8\n]", "But ( R(1) = a(1) + b = a + b ), so:", "[\na + b = 8 \quad \ ext{(Equation 1)}\n]", "---", "## Step 4: Compute Derivative ( f'(u) )", "Differentiate ( f(u) ):", "[\nf'(u) = 5u^4 - 9u^2 + 2\n]", "Evaluate at ( u = 1 ):", "[\nf'(1) = 5(1)^4 - 9(1)^2 + 2 = 5 - 9 + 2 = -2\n]", "Since ( R(u) = au + b ), ( R'(u) = a ), so:", "[\na = f'(1) = -2 \quad \ ext{(Equation 2)}\n]", "---", "## Step 5: Solve the System", "From Equation 2: ( a = -2 )\nSubstitute into Equation 1:", "[\n-2 + b = 8 \Rightarrow b = 10\n]", "---", "## Step 6: Write the Final Remainder", "[\nR(u) = -2u + 10\n]", "---", "## Conclusion", "When dividing ( u^5 - 3u^3 + 2u + 8 ) by ( u^2 - 2u + 1 = (u - 1)^2 ), the remainder is a linear polynomial. Using polynomial remainder techniques and differentiation at the double root, we found:", "[\n\boxed{-2u + 10}\n]", "This remainder is crucial for simplifying expressions, analyzing behavior near ( u = 1 ), and solving polynomial equations symbolically.", "---", "## Bonus Tip: Verify with Polynomial Division", "To confirm, long division or synthetic division approach (extended for quadratic divisors) yields the same result. For advanced learners, using polynomial remainder theorem generalized methods or resultant calculations also leads to the same remainder.", "---", "# Key SEO Keywords:\n- Remainder when dividing polynomials\n- ( u^5 - 3u^3 + 2u + 8 ) divided by ( u^2 - 2u + 1 )\n- Polynomial division remainder formula\n- Find remainder using derivative method\n- Double root polynomial remainder\n- Algebraic division with remainder (u−1)²", "---", "Improving your grasp of remainders in polynomial division empowers your algebra skills and supports advanced work in calculus, signal processing, and theoretical mathematics."]