{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0001","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [0]，求 p(-12) 的精確值。","answer":{"kind":"numeric","value":"0","explanation_zh":"p(-12)=0。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032001,"parameters":{"coefficients":[0],"x":-12,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[],\"operation\":\"evaluate\",\"x\":-12}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0002","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [5]，求 p(-11) 的精確值。","answer":{"kind":"numeric","value":"5","explanation_zh":"p(-11)=5。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032002,"parameters":{"coefficients":[5],"x":-11,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"5\"],\"operation\":\"evaluate\",\"x\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0003","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [0, 1]，求 p(-10) 的精確值。","answer":{"kind":"numeric","value":"-10","explanation_zh":"p(-10)=-10。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032003,"parameters":{"coefficients":[0,1],"x":-10,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"0\",\"1\"],\"operation\":\"evaluate\",\"x\":-10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0004","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-9, 0, -1]，求 p(-9) 的精確值。","answer":{"kind":"numeric","value":"-90","explanation_zh":"p(-9)=-90。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032004,"parameters":{"coefficients":[-9,0,-1],"x":-9,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-9\",\"0\",\"-1\"],\"operation\":\"evaluate\",\"x\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0005","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-8, 1]，求 p(-8) 的精確值。","answer":{"kind":"numeric","value":"-16","explanation_zh":"p(-8)=-16。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032005,"parameters":{"coefficients":[-8,1],"x":-8,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-8\",\"1\"],\"operation\":\"evaluate\",\"x\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0006","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-7, 2, 1]，求 p(-7) 的精確值。","answer":{"kind":"numeric","value":"28","explanation_zh":"p(-7)=28。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032006,"parameters":{"coefficients":[-7,2,1],"x":-7,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-7\",\"2\",\"1\"],\"operation\":\"evaluate\",\"x\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0007","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-6, 3, -1]，求 p(-6) 的精確值。","answer":{"kind":"numeric","value":"-60","explanation_zh":"p(-6)=-60。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032007,"parameters":{"coefficients":[-6,3,-1],"x":-6,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-6\",\"3\",\"-1\"],\"operation\":\"evaluate\",\"x\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0008","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-5, -3]，求 p(-5) 的精確值。","answer":{"kind":"numeric","value":"10","explanation_zh":"p(-5)=10。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032008,"parameters":{"coefficients":[-5,-3],"x":-5,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-5\",\"-3\"],\"operation\":\"evaluate\",\"x\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0009","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-4, -2, 1]，求 p(-4) 的精確值。","answer":{"kind":"numeric","value":"20","explanation_zh":"p(-4)=20。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032009,"parameters":{"coefficients":[-4,-2,1],"x":-4,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-4\",\"-2\",\"1\"],\"operation\":\"evaluate\",\"x\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0010","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-3, -1, -1]，求 p(-3) 的精確值。","answer":{"kind":"numeric","value":"-9","explanation_zh":"p(-3)=-9。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032010,"parameters":{"coefficients":[-3,-1,-1],"x":-3,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-3\",\"-1\",\"-1\"],\"operation\":\"evaluate\",\"x\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0011","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-2]，求 p(-2) 的精確值。","answer":{"kind":"numeric","value":"-2","explanation_zh":"p(-2)=-2。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032011,"parameters":{"coefficients":[-2],"x":-2,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-2\"],\"operation\":\"evaluate\",\"x\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0012","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [-1, 1, 1]，求 p(-1) 的精確值。","answer":{"kind":"numeric","value":"-1","explanation_zh":"p(-1)=-1。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032012,"parameters":{"coefficients":[-1,1,1],"x":-1,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-1\",\"1\",\"1\"],\"operation\":\"evaluate\",\"x\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0013","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [0, 2, -1]，求 p(0) 的精確值。","answer":{"kind":"numeric","value":"0","explanation_zh":"p(0)=0。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032013,"parameters":{"coefficients":[0,2,-1],"x":0,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"0\",\"2\",\"-1\"],\"operation\":\"evaluate\",\"x\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0014","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [1, 3]，求 p(1) 的精確值。","answer":{"kind":"numeric","value":"4","explanation_zh":"p(1)=4。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032014,"parameters":{"coefficients":[1,3],"x":1,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"1\",\"3\"],\"operation\":\"evaluate\",\"x\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0015","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [2, -3, 1]，求 p(2) 的精確值。","answer":{"kind":"numeric","value":"0","explanation_zh":"p(2)=0。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032015,"parameters":{"coefficients":[2,-3,1],"x":2,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"2\",\"-3\",\"1\"],\"operation\":\"evaluate\",\"x\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0016","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [3, -2, -1]，求 p(3) 的精確值。","answer":{"kind":"numeric","value":"-12","explanation_zh":"p(3)=-12。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032016,"parameters":{"coefficients":[3,-2,-1],"x":3,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"3\",\"-2\",\"-1\"],\"operation\":\"evaluate\",\"x\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0017","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [4, -1]，求 p(4) 的精確值。","answer":{"kind":"numeric","value":"0","explanation_zh":"p(4)=0。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032017,"parameters":{"coefficients":[4,-1],"x":4,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"4\",\"-1\"],\"operation\":\"evaluate\",\"x\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0018","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [5, 0, 1]，求 p(5) 的精確值。","answer":{"kind":"numeric","value":"30","explanation_zh":"p(5)=30。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032018,"parameters":{"coefficients":[5,0,1],"x":5,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"5\",\"0\",\"1\"],\"operation\":\"evaluate\",\"x\":5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0019","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [6, 1, -1]，求 p(6) 的精確值。","answer":{"kind":"numeric","value":"-24","explanation_zh":"p(6)=-24。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032019,"parameters":{"coefficients":[6,1,-1],"x":6,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"6\",\"1\",\"-1\"],\"operation\":\"evaluate\",\"x\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0020","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [7, 2]，求 p(7) 的精確值。","answer":{"kind":"numeric","value":"21","explanation_zh":"p(7)=21。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032020,"parameters":{"coefficients":[7,2],"x":7,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"7\",\"2\"],\"operation\":\"evaluate\",\"x\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0021","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [8, 3, 1]，求 p(8) 的精確值。","answer":{"kind":"numeric","value":"96","explanation_zh":"p(8)=96。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032021,"parameters":{"coefficients":[8,3,1],"x":8,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"8\",\"3\",\"1\"],\"operation\":\"evaluate\",\"x\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0022","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [9, -3, -1]，求 p(9) 的精確值。","answer":{"kind":"numeric","value":"-99","explanation_zh":"p(9)=-99。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032022,"parameters":{"coefficients":[9,-3,-1],"x":9,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"9\",\"-3\",\"-1\"],\"operation\":\"evaluate\",\"x\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0023","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [10, -2]，求 p(10) 的精確值。","answer":{"kind":"numeric","value":"-10","explanation_zh":"p(10)=-10。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032023,"parameters":{"coefficients":[10,-2],"x":10,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"10\",\"-2\"],\"operation\":\"evaluate\",\"x\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0024","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [11, -1, 1]，求 p(11) 的精確值。","answer":{"kind":"numeric","value":"121","explanation_zh":"p(11)=121。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032024,"parameters":{"coefficients":[11,-1,1],"x":11,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"11\",\"-1\",\"1\"],\"operation\":\"evaluate\",\"x\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0025","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式精確代入","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。p 的係數為 [12, 0, -1]，求 p(12) 的精確值。","answer":{"kind":"numeric","value":"-132","explanation_zh":"p(12)=-132。各係數乘上其對應的自然數次冪後相加；這是單點代入值，不是一般多項式恆等式證明。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式精確代入"]},"variant":{"family_id":"qf-polynomial-exact-evaluation","seed":202610032025,"parameters":{"coefficients":[12,0,-1],"x":12,"operation":"evaluate","semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"12\",\"0\",\"-1\"],\"operation\":\"evaluate\",\"x\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0026","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [0] 中，x^0 的係數 a_0 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_0=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032026,"parameters":{"coefficients":[0],"k":0,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[],\"k\":0,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0027","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-9, -1, 2] 中，x^1 的係數 a_1 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_1=-1。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032027,"parameters":{"coefficients":[-9,-1,2],"k":1,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-9\",\"-1\",\"2\"],\"k\":1,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0028","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-8, 0, 3] 中，x^2 的係數 a_2 是多少？","answer":{"kind":"numeric","value":"3","explanation_zh":"a_2=3。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032028,"parameters":{"coefficients":[-8,0,3],"k":2,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-8\",\"0\",\"3\"],\"k\":2,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0029","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-7, 1, 4] 中，x^3 的係數 a_3 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_3=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032029,"parameters":{"coefficients":[-7,1,4],"k":3,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-7\",\"1\",\"4\"],\"k\":3,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0030","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-6, 2, 1] 中，x^4 的係數 a_4 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032030,"parameters":{"coefficients":[-6,2,1],"k":4,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-6\",\"2\",\"1\"],\"k\":4,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0031","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-5, -2, 2] 中，x^0 的係數 a_0 是多少？","answer":{"kind":"numeric","value":"-5","explanation_zh":"a_0=-5。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032031,"parameters":{"coefficients":[-5,-2,2],"k":0,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-5\",\"-2\",\"2\"],\"k\":0,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0032","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-4, -1, 3] 中，x^1 的係數 a_1 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_1=-1。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032032,"parameters":{"coefficients":[-4,-1,3],"k":1,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-4\",\"-1\",\"3\"],\"k\":1,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0033","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-3, 0, 4] 中，x^2 的係數 a_2 是多少？","answer":{"kind":"numeric","value":"4","explanation_zh":"a_2=4。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032033,"parameters":{"coefficients":[-3,0,4],"k":2,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-3\",\"0\",\"4\"],\"k\":2,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0034","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-2, 1, 1] 中，x^3 的係數 a_3 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_3=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032034,"parameters":{"coefficients":[-2,1,1],"k":3,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-2\",\"1\",\"1\"],\"k\":3,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0035","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [-1, 2, 2] 中，x^4 的係數 a_4 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032035,"parameters":{"coefficients":[-1,2,2],"k":4,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-1\",\"2\",\"2\"],\"k\":4,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0039","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [3, 1, 2] 中，x^3 的係數 a_3 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_3=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032039,"parameters":{"coefficients":[3,1,2],"k":3,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"3\",\"1\",\"2\"],\"k\":3,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0040","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [4, 2, 3] 中，x^4 的係數 a_4 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032040,"parameters":{"coefficients":[4,2,3],"k":4,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"4\",\"2\",\"3\"],\"k\":4,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0041","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [5, -2, 4] 中，x^0 的係數 a_0 是多少？","answer":{"kind":"numeric","value":"5","explanation_zh":"a_0=5。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032041,"parameters":{"coefficients":[5,-2,4],"k":0,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"5\",\"-2\",\"4\"],\"k\":0,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0042","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [6, -1, 1] 中，x^1 的係數 a_1 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_1=-1。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032042,"parameters":{"coefficients":[6,-1,1],"k":1,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"6\",\"-1\",\"1\"],\"k\":1,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0043","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [7, 0, 2] 中，x^2 的係數 a_2 是多少？","answer":{"kind":"numeric","value":"2","explanation_zh":"a_2=2。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032043,"parameters":{"coefficients":[7,0,2],"k":2,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"7\",\"0\",\"2\"],\"k\":2,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0044","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [8, 1, 3] 中，x^3 的係數 a_3 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_3=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032044,"parameters":{"coefficients":[8,1,3],"k":3,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"8\",\"1\",\"3\"],\"k\":3,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0045","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [9, 2, 4] 中，x^4 的係數 a_4 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032045,"parameters":{"coefficients":[9,2,4],"k":4,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"9\",\"2\",\"4\"],\"k\":4,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0046","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [10, -2, 1] 中，x^0 的係數 a_0 是多少？","answer":{"kind":"numeric","value":"10","explanation_zh":"a_0=10。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032046,"parameters":{"coefficients":[10,-2,1],"k":0,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"10\",\"-2\",\"1\"],\"k\":0,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0047","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [11, -1, 2] 中，x^1 的係數 a_1 是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_1=-1。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032047,"parameters":{"coefficients":[11,-1,2],"k":1,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"11\",\"-1\",\"2\"],\"k\":1,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0048","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [12, 0, 3] 中，x^2 的係數 a_2 是多少？","answer":{"kind":"numeric","value":"3","explanation_zh":"a_2=3。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032048,"parameters":{"coefficients":[12,0,3],"k":2,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"12\",\"0\",\"3\"],\"k\":2,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0049","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [13, 1, 4] 中，x^3 的係數 a_3 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_3=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032049,"parameters":{"coefficients":[13,1,4],"k":3,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"13\",\"1\",\"4\"],\"k\":3,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0050","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"讀取指定次冪的係數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。在 [14, 2, 1] 中，x^4 的係數 a_4 是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。索引就是非負次冪；超出有限支撐的係數為零，不由字面項數推論。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["讀取指定次冪的係數"]},"variant":{"family_id":"qf-polynomial-coefficient","seed":202610032050,"parameters":{"coefficients":[14,2,1],"k":4,"operation":"coefficient","semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"14\",\"2\",\"1\"],\"k\":4,\"operation\":\"coefficient\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0051","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[0] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"未定義","explanation_zh":"所有係數皆為零，這是零多項式，次數未定義；不能按向量長度誤判為零次。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032051,"parameters":{"coefficients":[0],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0052","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-4] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"0","explanation_zh":"最大非零係數的索引為 0，所以次數是 0；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032052,"parameters":{"coefficients":[-4],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-4\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0053","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-10, 0, 1] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"2","explanation_zh":"最大非零係數的索引為 2，所以次數是 2；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032053,"parameters":{"coefficients":[-10,0,1],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-10\",\"0\",\"1\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0054","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-9, 0, 0, 1] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"3","explanation_zh":"最大非零係數的索引為 3，所以次數是 3；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032054,"parameters":{"coefficients":[-9,0,0,1],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-9\",\"0\",\"0\",\"1\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0055","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-8, 0, 0, 0, 1] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"4","explanation_zh":"最大非零係數的索引為 4，所以次數是 4；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032055,"parameters":{"coefficients":[-8,0,0,0,1],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-8\",\"0\",\"0\",\"0\",\"1\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0056","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[2] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"0","explanation_zh":"最大非零係數的索引為 0，所以次數是 0；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032056,"parameters":{"coefficients":[2],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"2\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0057","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-6, 2] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"1","explanation_zh":"最大非零係數的索引為 1，所以次數是 1；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032057,"parameters":{"coefficients":[-6,2],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-6\",\"2\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0058","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-5, 0, 2] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"2","explanation_zh":"最大非零係數的索引為 2，所以次數是 2；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032058,"parameters":{"coefficients":[-5,0,2],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-5\",\"0\",\"2\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0059","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-4, 0, 0, 2] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"3","explanation_zh":"最大非零係數的索引為 3，所以次數是 3；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032059,"parameters":{"coefficients":[-4,0,0,2],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-4\",\"0\",\"0\",\"2\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0060","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-3, 0, 0, 0, 2] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"4","explanation_zh":"最大非零係數的索引為 4，所以次數是 4；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032060,"parameters":{"coefficients":[-3,0,0,0,2],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-3\",\"0\",\"0\",\"0\",\"2\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0061","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[3] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"0","explanation_zh":"最大非零係數的索引為 0，所以次數是 0；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032061,"parameters":{"coefficients":[3],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"3\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0062","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[-1, 3] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"1","explanation_zh":"最大非零係數的索引為 1，所以次數是 1；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032062,"parameters":{"coefficients":[-1,3],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-1\",\"3\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0063","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[0, 0, 3] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"2","explanation_zh":"最大非零係數的索引為 2，所以次數是 2；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032063,"parameters":{"coefficients":[0,0,3],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"0\",\"0\",\"3\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0064","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[1, 0, 0, 3] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"3","explanation_zh":"最大非零係數的索引為 3，所以次數是 3；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032064,"parameters":{"coefficients":[1,0,0,3],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"1\",\"0\",\"0\",\"3\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0065","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[2, 0, 0, 0, 3] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"4","explanation_zh":"最大非零係數的索引為 4，所以次數是 4；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032065,"parameters":{"coefficients":[2,0,0,0,3],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"2\",\"0\",\"0\",\"0\",\"3\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0066","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[4] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"0","explanation_zh":"最大非零係數的索引為 0，所以次數是 0；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032066,"parameters":{"coefficients":[4],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"4\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0067","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[4, 4] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"1","explanation_zh":"最大非零係數的索引為 1，所以次數是 1；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032067,"parameters":{"coefficients":[4,4],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"4\",\"4\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0068","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[5, 0, 4] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"2","explanation_zh":"最大非零係數的索引為 2，所以次數是 2；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032068,"parameters":{"coefficients":[5,0,4],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"5\",\"0\",\"4\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0069","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[6, 0, 0, 4] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"3","explanation_zh":"最大非零係數的索引為 3，所以次數是 3；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032069,"parameters":{"coefficients":[6,0,0,4],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"6\",\"0\",\"0\",\"4\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0070","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[7, 0, 0, 0, 4] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"4","explanation_zh":"最大非零係數的索引為 4，所以次數是 4；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032070,"parameters":{"coefficients":[7,0,0,0,4],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"7\",\"0\",\"0\",\"0\",\"4\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0071","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[5] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"0","explanation_zh":"最大非零係數的索引為 0，所以次數是 0；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032071,"parameters":{"coefficients":[5],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"5\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0072","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[9, 5] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"1","explanation_zh":"最大非零係數的索引為 1，所以次數是 1；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032072,"parameters":{"coefficients":[9,5],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"9\",\"5\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0073","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[10, 0, 5] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"2","explanation_zh":"最大非零係數的索引為 2，所以次數是 2；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032073,"parameters":{"coefficients":[10,0,5],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"10\",\"0\",\"5\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0074","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[11, 0, 0, 5] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"3","explanation_zh":"最大非零係數的索引為 3，所以次數是 3；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032074,"parameters":{"coefficients":[11,0,0,5],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"11\",\"0\",\"0\",\"5\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0075","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式的次數","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。[12, 0, 0, 0, 5] 所表示多項式的次數為何？零多項式答「未定義」。","answer":{"kind":"short_text","value":"4","explanation_zh":"最大非零係數的索引為 4，所以次數是 4；尾端零不增加次數。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式的次數"]},"variant":{"family_id":"qf-polynomial-degree","seed":202610032075,"parameters":{"coefficients":[12,0,0,0,5],"operation":"degree","semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"12\",\"0\",\"0\",\"0\",\"5\"],\"operation\":\"degree\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0076","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [0] 與 [0] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0]","explanation_zh":"每個次數的係數分別相加，得到 [0]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032076,"parameters":{"left":[0],"right":[0],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[],\"operation\":\"add\",\"right\":[]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0077","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [2, 3] 與 [-2, -3] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0]","explanation_zh":"每個次數的係數分別相加，得到 [0]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032077,"parameters":{"left":[2,3],"right":[-2,-3],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-2\",\"-3\"],\"operation\":\"add\",\"right\":[\"2\",\"3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0078","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-10, 3] 與 [10, 1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 4]","explanation_zh":"每個次數的係數分別相加，得到 [0, 4]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032078,"parameters":{"left":[-10,3],"right":[10,1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-10\",\"3\"],\"operation\":\"add\",\"right\":[\"10\",\"1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0079","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-9, 4] 與 [9, -1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"每個次數的係數分別相加，得到 [0, 3]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032079,"parameters":{"left":[-9,4],"right":[9,-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-9\",\"4\"],\"operation\":\"add\",\"right\":[\"9\",\"-1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0080","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-8, 1] 與 [8] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 1]","explanation_zh":"每個次數的係數分別相加，得到 [0, 1]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032080,"parameters":{"left":[-8,1],"right":[8],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-8\",\"1\"],\"operation\":\"add\",\"right\":[\"8\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0081","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-7, 2] 與 [7, 1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"每個次數的係數分別相加，得到 [0, 3]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032081,"parameters":{"left":[-7,2],"right":[7,1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-7\",\"2\"],\"operation\":\"add\",\"right\":[\"7\",\"1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0082","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-6, 3] 與 [6, -1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 2]","explanation_zh":"每個次數的係數分別相加，得到 [0, 2]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032082,"parameters":{"left":[-6,3],"right":[6,-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-6\",\"3\"],\"operation\":\"add\",\"right\":[\"6\",\"-1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0083","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-5, 4] 與 [5] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 4]","explanation_zh":"每個次數的係數分別相加，得到 [0, 4]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032083,"parameters":{"left":[-5,4],"right":[5],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-5\",\"4\"],\"operation\":\"add\",\"right\":[\"5\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0084","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-4, 1] 與 [4, 1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 2]","explanation_zh":"每個次數的係數分別相加，得到 [0, 2]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032084,"parameters":{"left":[-4,1],"right":[4,1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-4\",\"1\"],\"operation\":\"add\",\"right\":[\"4\",\"1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0085","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-3, 2] 與 [3, -1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 1]","explanation_zh":"每個次數的係數分別相加，得到 [0, 1]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032085,"parameters":{"left":[-3,2],"right":[3,-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-3\",\"2\"],\"operation\":\"add\",\"right\":[\"3\",\"-1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0086","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-2, 3] 與 [2] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"每個次數的係數分別相加，得到 [0, 3]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032086,"parameters":{"left":[-2,3],"right":[2],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-2\",\"3\"],\"operation\":\"add\",\"right\":[\"2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0087","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [-1, 4] 與 [1, 1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 5]","explanation_zh":"每個次數的係數分別相加，得到 [0, 5]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032087,"parameters":{"left":[-1,4],"right":[1,1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-1\",\"4\"],\"operation\":\"add\",\"right\":[\"1\",\"1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0088","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [0, 1] 與 [0, -1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0]","explanation_zh":"每個次數的係數分別相加，得到 [0]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032088,"parameters":{"left":[0,1],"right":[0,-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"0\",\"-1\"],\"operation\":\"add\",\"right\":[\"0\",\"1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0089","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [1, 2] 與 [-1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 2]","explanation_zh":"每個次數的係數分別相加，得到 [0, 2]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032089,"parameters":{"left":[1,2],"right":[-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-1\"],\"operation\":\"add\",\"right\":[\"1\",\"2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0090","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [2, 3] 與 [-2, 1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 4]","explanation_zh":"每個次數的係數分別相加，得到 [0, 4]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032090,"parameters":{"left":[2,3],"right":[-2,1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-2\",\"1\"],\"operation\":\"add\",\"right\":[\"2\",\"3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0091","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [3, 4] 與 [-3, -1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"每個次數的係數分別相加，得到 [0, 3]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032091,"parameters":{"left":[3,4],"right":[-3,-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-3\",\"-1\"],\"operation\":\"add\",\"right\":[\"3\",\"4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0092","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [4, 1] 與 [-4] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 1]","explanation_zh":"每個次數的係數分別相加，得到 [0, 1]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032092,"parameters":{"left":[4,1],"right":[-4],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-4\"],\"operation\":\"add\",\"right\":[\"4\",\"1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0093","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [5, 2] 與 [-5, 1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"每個次數的係數分別相加，得到 [0, 3]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032093,"parameters":{"left":[5,2],"right":[-5,1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-5\",\"1\"],\"operation\":\"add\",\"right\":[\"5\",\"2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0094","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [6, 3] 與 [-6, -1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 2]","explanation_zh":"每個次數的係數分別相加，得到 [0, 2]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032094,"parameters":{"left":[6,3],"right":[-6,-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-6\",\"-1\"],\"operation\":\"add\",\"right\":[\"6\",\"3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0095","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [7, 4] 與 [-7] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 4]","explanation_zh":"每個次數的係數分別相加，得到 [0, 4]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032095,"parameters":{"left":[7,4],"right":[-7],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-7\"],\"operation\":\"add\",\"right\":[\"7\",\"4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0096","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [8, 1] 與 [-8, 1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 2]","explanation_zh":"每個次數的係數分別相加，得到 [0, 2]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032096,"parameters":{"left":[8,1],"right":[-8,1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-8\",\"1\"],\"operation\":\"add\",\"right\":[\"8\",\"1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0097","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [9, 2] 與 [-9, -1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 1]","explanation_zh":"每個次數的係數分別相加，得到 [0, 1]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032097,"parameters":{"left":[9,2],"right":[-9,-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-9\",\"-1\"],\"operation\":\"add\",\"right\":[\"9\",\"2\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0098","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [10, 3] 與 [-10] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 3]","explanation_zh":"每個次數的係數分別相加，得到 [0, 3]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032098,"parameters":{"left":[10,3],"right":[-10],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-10\"],\"operation\":\"add\",\"right\":[\"10\",\"3\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0099","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [11, 4] 與 [-11, 1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0, 5]","explanation_zh":"每個次數的係數分別相加，得到 [0, 5]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032099,"parameters":{"left":[11,4],"right":[-11,1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-11\",\"1\"],\"operation\":\"add\",\"right\":[\"11\",\"4\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r02-polynomial-0100","batch_id":"batch-20261003-r02-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式係數逐項相加","stem_zh":"本題 Q[x] 的整數係數向量由低次至高次排列。求 [12, 1] 與 [-12, -1] 的多項式和，以去尾端零的係數向量回答。","answer":{"kind":"expression","value":"[0]","explanation_zh":"每個次數的係數分別相加，得到 [0]。尾端零移除，零多項式統一表示為 [0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式係數逐項相加"]},"variant":{"family_id":"qf-polynomial-addition","seed":202610032100,"parameters":{"left":[12,1],"right":[-12,-1],"operation":"add","semantic_template":"qf-polynomial-addition/{\"left\":[\"-12\",\"-1\"],\"operation\":\"add\",\"right\":[\"12\",\"1\"]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}