Divide \( u^4 + 0u^3 + 0u^2 + 0u + 3 \) by \( u^2 - 2u + 2 \)
["# Polynomial Division: Divide ( u^4 + 3 ) by ( u^2 - 2u + 2 )", "Dividing polynomials is a fundamental operation in algebra that helps simplify complex expressions and solve equations. In this article, we explore the division of ( u^4 + 3 ) by ( u^2 - 2u + 2 ) step-by-step using long division. We also examine the result in terms of quotient and remainder, offering insights into the structure of the division and its applications.", "## Problem Statement", "We aim to divide:", "[\n\frac{u^4 + 0u^3 + 0u^2 + 0u + 3}{u^2 - 2u + 2}\n]", "Note that the numerator is simplified to ( u^4 + 3 ) since all intermediate coefficients are zero.", "---", "## Step 1: Set Up Long Division", "We write:", "[\nu^4 + 0u^3 + 0u^2 + 0u + 3 \div (u^2 - 2u + 2)\n]", "Arrange both dividend and divisor in descending powers of ( u ):", "- Dividend: ( u^4 + 0u^3 + 0u^2 + 0u + 3 )\n- Divisor: ( u^2 - 2u + 2 )", "---", "## Step 2: Divide Leading Terms", "Divide the leading term of the dividend ( u^4 ) by the leading term of the divisor ( u^2 ):", "[\nu^4 \div u^2 = u^2\n]", "So, the first term of the quotient is ( u^2 ).", "---", "## Step 3: Multiply and Subtract", "Multiply ( u^2 ) by the entire divisor:", "[\nu^2 \cdot (u^2 - 2u + 2) = u^4 - 2u^3 + 2u^2\n]", "Subtract this from the dividend:", "[\n(u^4 + 0u^3 + 0u^2 + 0u + 3) - (u^4 - 2u^3 + 2u^2) = 2u^3 - 2u^2 + 0u + 3\n]", "---", "## Step 4: Repeat the Process", "Now divide the new leading term ( 2u^3 ) by ( u^2 ):", "[\n2u^3 \div u^2 = 2u\n]", "Next term in the quotient: ( +2u )", "Multiply ( 2u ) by the divisor:", "[\n2u \cdot (u^2 - 2u + 2) = 2u^3 - 4u^2 + 4u\n]", "Subtract:", "[\n(2u^3 - 2u^2 + 0u + 3) - (2u^3 - 4u^2 + 4u) = (2u^3 - 2u^3) + (-2u^2 + 4u^2) + (0u - 4u) + 3 = 2u^2 - 4u + 3\n]", "---", "## Step 5: Final Division Step", "Divide ( 2u^2 ) by ( u^2 ):", "[\n2u^2 \div u^2 = 2\n]", "Next quotient term: ( +2 )", "Multiply:", "[\n2 \cdot (u^2 - 2u + 2) = 2u^2 - 4u + 4\n]", "Subtract:", "[\n(2u^2 - 4u + 3) - (2u^2 - 4u + 4) = 0u^2 + 0u - 1 = -1\n]", "Remainder is ( -1 ), and the degree of the remainder (0) is less than the degree of the divisor (2), so division stops.", "---", "## Step 6: Final Result", "Putting all terms together:", "- Quotient: ( u^2 + 2u + 2 )\n- Remainder: ( -1 )", "Thus,", "[\n\frac{u^4 + 3}{u^2 - 2u + 2} = u^2 + 2u + 2 - \frac{1}{u^2 - 2u + 2}\n]", "---", "## Additional Insights: Polynomial Division Outcomes", "When dividing ( u^4 + 3 ) by ( u^2 - 2u + 2 ), we see the result is a quadratic polynomial plus a proper rational remainder. The remainder ( -1 ) indicates that ( u^2 - 2u + 2 ) is not a factor — the polynomial ( u^4 + 3 ) has no roots in common with the divisor.", "---", "## Applications", "Polynomial division is valuable in:", "- Simplifying rational expressions\n- Solving polynomial equations via factorization\n- Analyzing system stability in control theory\n- Circuit and signal processing in engineering", "Using division steps like this ensures a clear path to understanding algebraic relationships.", "---", "## Summary", "- Dividend: ( u^4 + 3 )\n- Divisor: ( u^2 - 2u + 2 )\n- Quotient: ( u^2 + 2u + 2 )\n- Remainder: ( -1 )", "So,", "[\nu^4 + 3 = (u^2 - 2u + 2)(u^2 + 2u + 2) - 1\n]", "Mastering division techniques helps unlock deeper algebraic and analytical skills.", "---", "### Key Search Terms", "- Polynomial division explained\n- Divide ( u^4 + 3 ) by ( u^2 - 2u + 2 )\n- Long division of polynomials\n- Quotient and remainder after division\n- Simplify rational expressions algebraically", "---", "Whether you're studying algebra, preparing for exams, or working in applied mathematics, understanding division of polynomials is essential—and this guide offers a clear step-by-step solution."]
