To find the remainder when \( t^4 + 3t^2 + 2 \) is divided by \( t^2 + 1 \), we use polynomial long division or the remainder theorem. We express \( t^4 + 3t^2 + 2 \) in the form:

["Finding the Remainder When ( t^4 + 3t^2 + 2 ) is Divided by ( t^2 + 1 ): A Step-by-Step Guide", "When tasked with dividing a polynomial like ( t^4 + 3t^2 + 2 ) by ( t^2 + 1 ), many students wonder whether to apply polynomial long division or use the remainder theorem. While the remainder theorem works well for linear divisors, it’s not directly applicable here—since ( t^2 + 1 ) is quadratic. However, polynomial long division remains a reliable method, and we can also streamline the process using a strategic algebraic approach. In this article, we’ll explore all three routes—long division, factoring with the remainder theorem insight, and partial polynomial decomposition—to show how to efficiently find the remainder when ( t^4 + 3t^2 + 2 ) is divided by ( t^2 + 1 ).", "---", "### Why Polynomial Long Division Works", "Polynomial long division mirrors the process of numerical long division, dividing the dividend polynomial ( t^4 + 3t^2 + 2 ) by the divisor ( t^2 + 1 ) step by step. Though slightly lengthy, this method guarantees an exact quotient and remainder of lower degree.", "Step 1: Divide the leading term: ( t^4 \div t^2 = t^2 ).\nMultiply ( t^2(t^2 + 1) = t^4 + t^2 ).\nSubtract: ( (t^4 + 3t^2 + 2) - (t^4 + t^2) = 2t^2 + 2 ).", "Step 2: Divide ( 2t^2 \div t^2 = 2 ).\nMultiply ( 2(t^2 + 1) = 2t^2 + 2 ).\nSubtract: ( (2t^2 + 2) - (2t^2 + 2) = 0 ).", "So, the division yields:\n[\nt^4 + 3t^2 + 2 = (t^2 + 1)(t^2 + 2) + 0\n]", "Wait — the remainder is 0? Not quite. Let’s examine this result carefully.", "Actually, we observe:\n[\n t^4 + 3t^2 + 2 = (t^4 + t^2) + (2t^2 + 2) = (t^2)(t^2 + 1) + 2(t^2 + 1) = (t^2 + 2)(t^2 + 1)\n]\nHence,\n[\nt^4 + 3t^2 + 2 = (t^2 + 1)(t^2 + 2)\n]\nThis shows that ( t^2 + 1 ) is a factor of the polynomial — meaning the division is exact, and the remainder is 0.", "---", "### Understanding the Remainder via Polynomial Division", "The remainder after dividing any polynomial ( f(t) ) by a divisor ( d(t) ) of degree 2 must be of degree less than 2 — so it’s of the form ( r(t) = at + b ). In this case, we found this remainder to be zero, confirming divisibility.", "Alternatively, if the division weren’t exact, we would write:\n[\nt^4 + 3t^2 + 2 = (t^2 + 1) \cdot Q(t) + at + b\n]\nBut since the division yields no remainder, ( at + b = 0 \Rightarrow a = 0, b = 0 ).", "---", "### Using the Remainder Theorem (Extended Idea)", "Although the classical remainder theorem applies to linear divisors, we can adapt its logic. For a divisor of the form ( t^2 + 1 ), note that its roots are ( t = i ) and ( t = -i ). By the polynomial remainder theorem generalized to higher degrees, the remainder upon dividing by ( t^2 + 1 ) is a linear polynomial ( r(t) = at + b ), evaluated such that:\n[\nf(t) = (t^2 + 1) Q(t) + at + b\n]\nSubstituting ( t = i ):\n[\nf(i) = (i^2 + 1)Q(i) + ai + b = 0 + ai + b = ai + b\n]\nBut ( f(i) = i^4 + 3i^2 + 2 = 1 + 3(-1) + 2 = 1 - 3 + 2 = 0 )\nSo:\n[\nai + b = 0 \Rightarrow a = 0, b = 0\n]\nSimilarly, ( f(-i) = 0 \Rightarrow -ai + b = 0 \Rightarrow a = 0, b = 0 )\nThus, the remainder is ( 0 ).", "This confirms our long division result through algebraic evaluation of complex roots.", "---", "### Alternative: Factoring as a Shortcut", "Because ( t^2 + 1 ) divides evenly, we can factor:\nLet’s write:\n[\nt^4 + 3t^2 + 2 = (t^2 + 1)(t^2 + 2)\n]\nThis factors completely as:\n[\n(t^2 + 1)(t^2 + 2) = t^4 + 2t^2 + t^2 + 2 = t^4 + 3t^2 + 2\n]\nMatching perfectly — confirming no remainder exists.", "---", "### Final Conclusion: The Remainder is Zero", "Using polynomial long division, factoring, and extended remainder theorem logic, we conclude:", "[\n\ ext{Remainder of } \dfrac{t^4 + 3t^2 + 2}{t^2 + 1} = 0\n]", "This result tells us that ( t^2 + 1 ) is a factor of ( t^4 + 3t^2 + 2 ), and the division is exact. Understanding this process builds strong polynomial manipulation skills useful across algebra, calculus, and engineering applications.", "---", "Key Takeaways:\n- For divisors of degree ≥ 2, polynomial long division is reliable.\n- The remainder theorem generalizes: substitute roots of the divisor to find constant remainder.\n- Factoring and algebraic verification streamline the process when possible.\n- Precision in arithmetic and polynomial manipulation ensures accurate results.", "Mastering these techniques empowers accurate polynomial division — critical in symbolic computation, function analysis, and higher mathematics.", "---", "ried in ( t^4 + 3t^2 + 2 ) has a remainder of 0 when divided by ( t^2 + 1 ). Use long division, root substitution, or factoring to find the quotient and confirm exact divisibility."]