f(1) = 1^4 - 4 \cdot 1^3 + 6 \cdot 1^2 - 4 \cdot 1 + 1 = 1 - 4 + 6 - 4 + 1 = 0
["Understanding f(1) = 1⁴ - 4·1³ + 6·1² - 4·1 + 1: Why This Polynomial Always Equals Zero", "Have you ever wondered why a seemingly simple polynomial expression always evaluates to zero when ( x = 1 )? Let’s explore:\n[\nf(x) = x^4 - 4x^3 + 6x^2 - 4x + 1\n]\nAt first glance, plugging in ( x = 1 ) appears to yield:\n[\n1 - 4 + 6 - 4 + 1 = 0\n]\nBut what’s the deeper meaning behind this result? More importantly, why does this polynomial always equal zero when evaluated at ( x = 1 )?", "---", "### Step-by-Step Evaluation of f(1)", "Substitute ( x = 1 ) directly into the polynomial:\n[\nf(1) = (1)^4 - 4(1)^3 + 6(1)^2 - 4(1) + 1\n]\nCalculate powers and products:\n[\n1 - 4 + 6 - 4 + 1\n]\nNow simplify step-by-step:\n[\n1 - 4 = -3\n]\n[\n-3 + 6 = 3\n]\n[\n3 - 4 = -1\n]\n[\n-1 + 1 = 0\n]\nThus, indeed,\n[\nf(1) = 0\n]\nBut why?", "---", "### The Pattern: Binomial Coefficients and Expansion", "Observe that the coefficients ( 1, -4, 6, -4, 1 ) strongly resemble the binomial expansion of ( (x - 1)^4 ).", "Recall the Binomial Theorem:\n[\n(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\n]", "Apply this with ( a = x ), ( b = -1 ), and ( n = 4 ):\n[\n(x - 1)^4 = \binom{4}{0}x^4(-1)^0 + \binom{4}{1}x^3(-1)^1 + \binom{4}{2}x^2(-1)^2 + \binom{4}{3}x^1(-1)^3 + \binom{4}{4}x^0(-1)^4\n]\n[\n= 1 \cdot x^4 - 4x^3 + 6x^2 - 4x + 1\n]\nWhich matches exactly:\n[\nf(x) = (x - 1)^4\n]", "---", "### Why f(1) = 0: A Special Root", "Because:\n[\nf(x) = (x - 1)^4\n]\nevaluating at ( x = 1 ) gives:\n[\nf(1) = (1 - 1)^4 = 0^4 = 0\n]\nMore importantly, ( x = 1 ) is a root of multiplicity 4 — the function touches the x-axis at ( x = 1 ) but curves flatten due to the high power.", "This explains why every time we compute ( f(1) ), the result is always zero — it’s not a coincidence, but a structural property of the polynomial’s form.", "---", "### Significance in Algebra and Beyond", "Polynomials like ( f(x) = (x - 1)^4 ) are fundamental in algebra because:", "- They represent repeated roots, crucial for graph behavior (e.g., flatness at roots with high multiplicity).\n- They appear in Taylor expansions and approximations (the 4th power reflects the fourth-order behavior in calculus).\n- Such polynomials model real-world phenomena with stable equilibrium points, such as in physics, economics, and engineering.", "---", "### Conclusion", "The evaluation ( f(1) = 0 ) is not accidental—it follows directly from the polynomial’s identity as ( (x - 1)^4 ). Recognizing this connection illuminates deeper principles in algebra: the power of binomial expansions and the meaningful behavior of polynomials at specific values.", "So next time you compute ( f(1) ), remember: you’re not just calculating a number—you’re unlocking a fundamental truth in polynomial mathematics.", "---", "Keywords: \nPolynomialEvaluation #f(1) = 0 #MathExplanation #BinomialExpansion #(x - 1)^4 #Algebra #PolynomialRoots #CalculusFundamentals", "Meta Description:\nDiscover why ( f(1) = 1^4 - 4 \cdot 1^3 + 6 \cdot 1^2 - 4 \cdot 1 + 1 ) always equals zero. Learn how this polynomial factores as ( (x - 1)^4 ), explaining its essential root and algebraic structure."]
