{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-factorization",
  "semantic_family_id": "sf-factorization",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有理係數多項式因式分解"
  },
  "summary": {
    "zh-Hant": "因式分解是在指定係數域中把多項式寫成因子乘積；驗收比較全部係數而不是若干代入值。"
  },
  "statement": {
    "zh-Hant": "在 Q[x] 中，p 的因式分解候選 p=f·g 是多項式恆等式：展開並合併同次項後每個係數都相同。本條目示範二項式乘積、提公因子與平方差，不宣稱提供全部多項式的完整分解演算法。"
  },
  "formula": {
    "tex": "x^{2}-a^{2}",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></math>",
    "semantic_ast": {
      "type": "subtraction",
      "left": {
        "type": "power",
        "base": {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        "exponent": {
          "type": "number",
          "value": 2
        }
      },
      "right": {
        "type": "power",
        "base": {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        },
        "exponent": {
          "type": "number",
          "value": 2
        }
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "6887c5a4f89b4c31dc61a2c8976245d1d3cf7f061965653e5a766057e6aa1ac2"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "固定一個形式變數 x，係數在 Q 中；以有限係數資料表示。"
    },
    {
      "id": "a2",
      "zh-Hant": "因子及乘積都先去尾端零；本輪不把乘常數或重排因子的表示差異當成新的数学內容。"
    }
  ],
  "symbols": [
    {
      "token": "x",
      "role": "indeterminate",
      "role_zh": "形式變數",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "rational_constant",
      "role_zh": "代表平方差的有理常數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-polynomial",
      "title_zh": "一元有理係數多項式",
      "reason_zh": "係數比較、乘法卷積與恆等式在同一個多項式域中定義。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "(x+a)(x+b)=x²+(a+b)x+ab；逐項乘法與合併同次項提供可重算的係數身份。對 a∈Q，提公因子例如 ax+a=a(x+1)，仍在同一個一元多項式域中。",
      "平方差 x²−d²=(x−d)(x+d) 是恆等式。本輪用整數 d 及一元係數展開驗證；不由一兩個相同代入值推出完整多項式相等。",
      "可分解性取決於係數域。例如 x²−2 在 Q 中不能分成兩個非定常一次因子，在 R 中卻可使用 ±√2。因式分解不是方程式求根，使用零積性求解還需要指定數域。"
    ],
    "common_misconception_zh": "把平方和套用平方差公式，或用有限數值吻合冒充沒有次數界證據的多項式恆等式。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只驗收明示有限係數的乘積身份；未宣稱一般不可約性、唯一分解或完整演算法已形式證明。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定 Polynomial Q 乘法、coeff_mul、指定恆等式及係數域轉換條件。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-factorization.md 整理；2026-10-03 脈衝 R03，英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}