Question:** A mathematician working on algebraic topology defines a function \( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 \). Determine the remainder when \( f(x) \) is divided by \( x - 1 \).

["Mastering Polynomial Division: Finding the Remainder of ( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 ) Divided by ( x - 1 )", "In the world of algebraic topology and pure mathematics, understanding how functions behave under division is crucial—especially when evaluating polynomials. One fundamental concept is the Remainder Theorem, a powerful tool that simplifies finding remainders without performing full polynomial long division.", "### Understanding the Remainder Theorem", "The Remainder Theorem states that if a polynomial ( f(x) ) is divided by ( x - a ), the remainder of that division is simply ( f(a) ).", "This elegant result eliminates the need for lengthy synthetic or long division, offering a quick and efficient solution—perfect for both classroom learning and advanced applications in topology and geometry.", "### Applying the Remainder Theorem to the Given Function", "We are given the polynomial:\n[\nf(x) = x^4 - 4x^3 + 6x^2 - 4x + 1\n]", "We are to find the remainder when ( f(x) ) is divided by ( x - 1 ), so we set ( a = 1 ) and compute ( f(1) ).", "[\nf(1) = (1)^4 - 4(1)^3 + 6(1)^2 - 4(1) + 1\n]", "Evaluate step by step:", "- ( 1^4 = 1 )\n- ( -4 \cdot 1^3 = -4 )\n- ( 6 \cdot 1^2 = 6 )\n- ( -4 \cdot 1 = -4 )\n- Constant term: ( +1 )", "Add them together:", "[\n1 - 4 + 6 - 4 + 1 = (1 - 4) + (6 - 4) + 1 = (-3) + 2 + 1 = 0\n]", "Thus,\n[\nf(1) = 0\n]", "### Conclusion: The Remainder is Zero", "Since ( f(1) = 0 ), the remainder when ( f(x) ) is divided by ( x - 1 ) is 0. This means ( x - 1 ) is a factor of ( f(x) ), a result that aligns beautifully with algebraic topology’s emphasis on structure and symmetry in polynomial forms.", "### Final Answer", "The remainder when ( f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 ) is divided by ( x - 1 ) is 0.", "---", "This application of the Remainder Theorem not only solves the immediate problem efficiently but also reinforces key concepts used in higher mathematics, including algebraic topology, where identifying roots and factors supports deeper exploration of topological spaces linked to polynomial invariants.", "For efficient computation and conceptual clarity, always apply the Remainder Theorem—especially when working with elegant polynomials like this one, which simplifies neatly to ( (x - 1)^4 ), as verified by expansion:", "[\nf(x) = (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1\n]", "This confirms the factorization and reaffirms that ( x - 1 ) divides ( f(x) ) perfectly."]