The remainder is \(-1\). Therefore, the remainder when \( x^4 + 3x^2 + 1 \) is divided by \( x^2 + 1 \) is \(\boxed{-1}\).
["Understanding Polynomial Division: Why the Remainder is (-1) When Dividing (x^4 + 3x^2 + 1) by (x^2 + 1)", "When dividing polynomials, the remainder plays a crucial role in polynomial division—just like in number division, where (17 \div 5 = 3) with a remainder of (2). In algebra, the remainder theorem for polynomials tells us that when dividing a polynomial (f(x)) by a divisor (d(x)), the remainder (R) satisfies (f(x) = q(x) \cdot d(x) + R), where degree of (R) is less than degree of (d(x)).", "In this article, we explore a classic example: what is the remainder when (f(x) = x^4 + 3x^2 + 1) is divided by (d(x) = x^2 + 1)? The mathematical insight reveals that the remainder is (\boxed{-1}). Let’s understand why.", "---", "### Polynomial Division Step-by-Step", "Let’s divide (f(x) = x^4 + 0x^3 + 3x^2 + 0x + 1) by (d(x) = x^2 + 1).", "Step 1: Divide leading terms\n(x^4 \div x^2 = x^2). Multiply (x^2) by (x^2 + 1):\n[\nx^2(x^2 + 1) = x^4 + x^2\n]\nSubtract from (f(x)):\n[\n(x^4 + 0x^3 + 3x^2 + 0x + 1) - (x^4 + x^2) = 0x^4 + 0x^3 + 2x^2 + 0x + 1\n]", "Step 2: Divide new leading term\nNow divide (2x^2 \div x^2 = 2). Multiply (2(x^2 + 1) = 2x^2 + 2).\nSubtract:\n[\n(2x^2 + 0x + 1) - (2x^2 + 2) = 0x^2 + 0x - 1\n]", "Now the remainder is (-1), and its degree ((0)) is less than the divisor’s degree ((2).", "Thus, we conclude:", "[\nx^4 + 3x^2 + 1 = (x^2 + 1)(x^2 + 2) - 1\n]", "---", "### Why the Remainder is (-1)", "This confirms that when dividing by (x^2 + 1), the remainder must be of lower degree, a polynomial of degree less than 2—so linear or constant. Here, it evaluates to a constant: (-1).", "The negative sign arises naturally from subtracting the final correction (2(x^2 + 1)) from the intermediate remainder (2x^2 + 0x + 1):", "[\n2x^2 + 1 - 2x^2 - 2 = -1\n]", "---", "### Why This Matters in Algebra and Beyond", "Understanding remainders in polynomial division is essential for:", "- Solving polynomial equations efficiently\n- Simplifying rational expressions\n- Analyzing roots and factorizations\n- Applications in coding theory, cryptography, and control systems", "In math competitions and advanced algebra courses, recognizing that remainders are always unique and lower-degree is key—especially when using division algorithms like polynomial long division or synthetic division for polynomials.", "---", "### Final Answer", "The remainder when (x^4 + 3x^2 + 1) is divided by (x^2 + 1) is:\n[\n\boxed{-1}\n]", "This result confirms the algebraic principle that division yields one unique quotient and a remainder of strictly smaller degree.", "---", "Keywords: polynomial division, remainder theorem for polynomials, divide (x^4 + 3x^2 + 1) by (x^2 + 1), remainder is (-1), math explanation, polynomial long division, algebra tutorial."]
