To find the remainder when \( x^4 + 3x^2 + 1 \) is divided by \( x^2 + 1 \), we perform polynomial long division.

["Finding the Remainder When ( x^4 + 3x^2 + 1 ) is Divided by ( x^2 + 1 ) Using Polynomial Long Division", "When dividing polynomials, one fundamental task is finding the remainder, especially when applying division algorithms in algebra. This article explains step-by-step how to find the remainder when dividing ( f(x) = x^4 + 3x^2 + 1 ) by ( d(x) = x^2 + 1 ) using polynomial long division—a clear, systematic method that reveals both quotient and remainder.", "---", "### Why Use Polynomial Long Division?\nPolynomial long division mirrors the arithmetic of numerical division but extends it to expressions involving variables. This technique allows us to break down complex polynomials into a quotient and a remainder where the degree of the remainder is always less than the degree of the divisor. Since ( d(x) = x^2 + 1 ) is degree 2, the remainder will be a polynomial of degree 1 or less—typically written as ( ax + b ).", "---", "### Step 1: Set Up the Division\nWe divide:\n[\nf(x) = x^4 + 3x^2 + 1 \quad \ ext{by} \quad d(x) = x^2 + 1\n]\nArrange both polynomials in descending powers of ( x ):\n- Dividend: ( x^4 + 0x^3 + 3x^2 + 0x + 1 )\n- Divisor: ( x^2 + 0x + 1 )", "---", "### Step 2: Divide Leading Terms\nStart by dividing the leading term of the dividend by the leading term of the divisor:\n[\n\frac{x^4}{x^2} = x^2\n]\nThis ( x^2 ) is the first term of the quotient.", "Multiply the entire divisor ( x^2 + 1 ) by ( x^2 ):\n[\nx^2(x^2 + 1) = x^4 + x^2\n]\nSubtract this product from the dividend:\n[\n(x^4 + 0x^3 + 3x^2 + 0x + 1) - (x^4 + 0x^3 + x^2) = 0x^4 + 0x^3 + 2x^2 + 0x + 1\n]\nResult: ( 2x^2 + 0x + 1 )", "---", "### Step 3: Repeat the Process\nNow divide the new leading term ( 2x^2 ) by ( x^2 ):\n[\n\frac{2x^2}{x^2} = 2\n]\nAdd this to the quotient: total quotient so far is ( x^2 + 2 ).", "Multiply the divisor ( x^2 + 1 ) by 2:\n[\n2(x^2 + 1) = 2x^2 + 2\n]\nSubtract:\n[\n(2x^2 + 0x + 1) - (2x^2 + 0x + 2) = 0x^2 + 0x - 1\n]\nRemainder: ( -1 ), which has degree 0—less than the degree 2 of the divisor.", "---", "### Final Result\nThe division yields:\n[\nx^4 + 3x^2 + 1 = (x^2 + 1)(x^2 + 2) + (-1)\n]\nThus, the remainder is ( -1 ).", "---", "### Key Takeaways\n- Polynomial long division is essential for finding remainders when dividing by quadratics or higher-degree polynomials.\n- The remainder always has a degree lower than the divisor.\n- Using the step-by-step subtraction and multiplication ensures accuracy.", "This method is valuable in algebra, engineering, and calculus—especially when simplifying rational functions or solving equations involving polynomial identities.", "---", "### Bonus: Verification\nPlug in a value for ( x ) to verify. Let ( x = 1 ):\n- LHS: ( 1^4 + 3(1)^2 + 1 = 1 + 3 + 1 = 5 )\n- RHS: ( (1^2 + 1)(1^2 + 2) - 1 = (2)(3) - 1 = 6 - 1 = 5 )\nMatch confirmed—remainder is correct.", "---", "Understanding polynomial division not only solves remainder problems but strengthens clarity in manipulating algebraic expressions. Use polynomial long division confidently—step by step, remainder guaranteed.", "Keywords: polynomial division, remainder with ( x^4 + 3x^2 + 1 )/( x^2 + 1 ), long division steps, algebraic remainder, degree of remainder, algebra practice."]