Find the remainder when \( u^4 + 3 \) is divided by \( u^2 - 2u + 2 \).
["# Find the Remainder When ( u^4 + 3 ) is Divided by ( u^2 - 2u + 2 )", "When dividing polynomials, finding the remainder is essential for simplifying expressions and solving equations. This article guides you step-by-step through finding the remainder when ( u^4 + 3 ) is divided by ( u^2 - 2u + 2 )—a crucial technique useful in algebra, calculus, and polynomial division applications.", "## Understanding Polynomial Division", "Dividing a polynomial ( f(u) ) by a divisor ( d(u) ) always yields a quotient ( q(u) ) and a remainder ( r(u) ) such that:\n[\nf(u) = d(u) \cdot q(u) + r(u)\n]\nwhere the degree of ( r(u) ) is less than the degree of ( d(u) ).\nIn this case, ( f(u) = u^4 + 3 ) and ( d(u) = u^2 - 2u + 2 ), a degree 2 polynomial. The remainder ( r(u) ) will therefore be at most degree 1 (i.e., ( r(u) = au + b )).", "---", "## Step-by-Step Solution", "We perform polynomial long division to divide ( u^4 + 3 ) by ( u^2 - 2u + 2 ).", "### Step 1: Divide leading terms\nDivide the leading term of the dividend, ( u^4 ), by the leading term of the divisor, ( u^2 ):\n[\n\frac{u^4}{u^2} = u^2\n]\nMultiply the entire divisor by ( u^2 ):\n[\nu^2(u^2 - 2u + 2) = u^4 - 2u^3 + 2u^2\n]\nSubtract from the original polynomial:\n[\n(u^4 + 0u^3 + 0u^2 + 0u + 3) - (u^4 - 2u^3 + 2u^2) = 2u^3 - 2u^2 + 0u + 3\n]", "---", "### Step 2: Repeat the process\nNow divide the new leading term ( 2u^3 ) by ( u^2 ):\n[\n\frac{2u^3}{u^2} = 2u\n]\nMultiply:\n[\n2u(u^2 - 2u + 2) = 2u^3 - 4u^2 + 4u\n]\nSubtract:\n[\n(2u^3 - 2u^2 + 0u + 3) - (2u^3 - 4u^2 + 4u) = (2u^2 - 4u + 3)\n]", "---", "### Step 3: Final division\nNow divide ( 2u^2 ) by ( u^2 ):\n[\n\frac{2u^2}{u^2} = 2\n]\nMultiply:\n[\n2(u^2 - 2u + 2) = 2u^2 - 4u + 4\n]\nSubtract:\n[\n(2u^2 - 4u + 3) - (2u^2 - 4u + 4) = -1\n]", "---", "## Final Result", "The division yields:\n[\nu^4 + 3 = (u^2 - 2u + 2)(u^2 + 2u + 2) - 1\n]\nThus, the remainder is:\n[\n\boxed{-1}\n]", "---", "## Summary", "Finding the remainder of ( u^4 + 3 ) divided by ( u^2 - 2u + 2 ) gives a constant value, ( -1 )—a concise result confirming the divisibility structure. Mastering polynomial remainder techniques sharpens algebraic skills and supports advanced mathematical computations.", "---", "### Additional Tips", "- Use polynomial long division for irreducible quadratic divisors.\n- Verify results by substituting values of ( u ) that satisfy the divisor’s roots (e.g., complex roots of ( u^2 - 2u + 2 = 0 )).\n- This remainder is key in simplifying complex rational functions or solving polynomial equations.", "---", "Keywords: remainder when ( u^4 + 3 ) is divided by ( u^2 - 2u + 2 ), polynomial division, remainder theorem, algebraic simplification, u^4 + 3 remainder, divisor ( u^2 - 2u + 2 ), complex arithmetic, division algorithm polynomials.", "---", "If you’re studying polynomial algebra or preparing for higher-level math, mastering division remainders is both practical and foundational. Let ( u^4 + 3 \div u^2 - 2u + 2 ) be a classic example of how structure reveals clarity in expressions!"]
