p(1) = 1^4 - 4 imes 1^3 + 6 imes 1^2 - 4 imes 1 + 1 = 1 - 4 + 6 - 4 + 1 = 0

["The Power of Polynomials: Proving p(1) = 1⁴ – 4×1³ + 6×1² – 4×1 + 1 Equals Zero", "In the world of algebra, polygons and polynomials come together in a beautiful and meaningful way that reveals deep mathematical truths. One of the most insightful examples is recognizing how the polynomial\np(1) = 1⁴ – 4×1³ + 6×1² – 4×1 + 1\nevaluates to zero — a result that embodies the heart of both calculus and combinatorics.", "### What is p(1) and Why Does It Matter?", "When we say p(1), we mean evaluating the polynomial at x = 1. This simple substitution turns a sequence of terms into a numeric value, often unlocking save-the-day insights like checking identities, factoring, or applying the Remainder Theorem.", "Here,\n[\np(x) = x^4 - 4x^3 + 6x^2 - 4x + 1\n]\nand substituting x = 1 gives:", "[\np(1) = 1^4 - 4(1)^3 + 6(1)^2 - 4(1) + 1 = 1 - 4 + 6 - 4 + 1\n]", "### A Closer Look: Simplifying the Expression", "Let’s compute step by step:\n- (1^4 = 1)\n- ( -4×1^3 = -4×1 = -4)\n- ( +6×1^2 = +6×1 = +6)\n- ( -4×1 = -4)\n- ( +1 = +1)", "Adding:\n[\n1 - 4 = -3,\quad -3 + 6 = 3,\quad 3 - 4 = -1,\quad -1 + 1 = 0\n]", "Thus,\n[\np(1) = 0\n]", "### So What Does p(1) = 0 Mean?", "This result is far from trivial. It signals that x = 1 is a root of the polynomial — a solution to the equation (p(x) = 0). In fact, this polynomial is more than just a number at (x=1); it reflects the elegant symmetry of binomial coefficients.", "The coefficients\n1, –4, 6, –4, 1\nare exactly the binomial expansion of ((x - 1)^4):", "[\n(x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1\n]", "So,\n[\np(x) = (x - 1)^4\n]", "This factorization clearly shows that (x = 1) is a root of multiplicity 4 — a repeated root — which explains why evaluating at (x=1) produces zero.", "### Real-World Implications", "Understanding p(1) = 0 helps in multiple domains:", "- Algebra: Verifying polynomial equations and factoring efficiently.\n- Calculus: Applying the Remainder Theorem, which states (p(a) = 0) means (x – a) is a factor of (p(x)).\n- Combinatorics: The expansion ((x - 1)^4) connects to counting combinations or distributions in probability and discrete math.", "### Recap: Why This Polynomial Stands Out", "This particular polynomial is a perfect example of how mathematics distills complexity into simplicity. By substituting (x = 1), we uncover:\n- A clear route to zero via simple arithmetic.\n- Full factorization using binomial theory.\n- Connection to repeated roots and higher polynomial structure.", "Whether you’re a student learning algebra, a teacher explaining key concepts, or simply a math enthusiast, recognizing that\n[\np(1) = 0\n]\nopens a window into powerful algebraic tools — rooted in symmetry, pattern recognition, and deep mathematical truth.", "---", "Key Takeaway:\nEvaluating polynomials at specific values like x = 1 is a fundamental technique that reveals hidden roots and structures. The polynomial ( (x - 1)^4 ) not only satisfies ( p(1) = 0 ), but also exemplifies how polynomials mirror natural patterns — from geometry to probability. Mastery of such ideas empowers deeper exploration across mathematics.", "Keywords: p(1) = 1⁴ – 4×1³ + 6×1² – 4×1 + 1, polynomial evaluation, binomial theorem, Remainder Theorem, (x – 1)⁴, algebraic identities, polynomial roots, combinatorics, math education."]