{"schema_version":"ocme-question-v0.1","id":"q-20261003-r03-factorization-0001","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [0] 與 [7, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[0]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [0]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033001,"parameters":{"left":[0],"right":[7,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"7\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0002","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-11, 1] 與 [3, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-33, -8, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-33, -8, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033002,"parameters":{"left":[-11,1],"right":[3,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-11\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0003","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-10, 1] 與 [4, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-40, -6, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-40, -6, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033003,"parameters":{"left":[-10,1],"right":[4,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-10\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0004","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-9, 1] 與 [5, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-45, -4, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-45, -4, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033004,"parameters":{"left":[-9,1],"right":[5,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-9\",\"1\"],\"operation\":\"product\",\"right\":[\"5\",\"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-r03-factorization-0005","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-8, 1] 與 [6, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-48, -2, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-48, -2, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033005,"parameters":{"left":[-8,1],"right":[6,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-8\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0006","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-7, 1] 與 [7, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-49, 0, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-49, 0, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033006,"parameters":{"left":[-7,1],"right":[7,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-7\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0007","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-6, 1] 與 [8, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-48, 2, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-48, 2, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033007,"parameters":{"left":[-6,1],"right":[8,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-6\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0008","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-5, 1] 與 [9, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-45, 4, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-45, 4, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033008,"parameters":{"left":[-5,1],"right":[9,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-5\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0009","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-4, 1] 與 [10, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-40, 6, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-40, 6, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033009,"parameters":{"left":[-4,1],"right":[10,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-4\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0010","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-3, 1] 與 [11, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-33, 8, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-33, 8, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033010,"parameters":{"left":[-3,1],"right":[11,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-3\",\"1\"],\"operation\":\"product\",\"right\":[\"11\",\"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-r03-factorization-0011","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-2, 1] 與 [12, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-24, 10, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-24, 10, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033011,"parameters":{"left":[-2,1],"right":[12,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-2\",\"1\"],\"operation\":\"product\",\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r03-factorization-0012","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-1, 1] 與 [13, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[-13, 12, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [-13, 12, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033012,"parameters":{"left":[-1,1],"right":[13,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"-1\",\"1\"],\"operation\":\"product\",\"right\":[\"13\",\"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-r03-factorization-0013","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [0, 1] 與 [14, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[0, 14, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [0, 14, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033013,"parameters":{"left":[0,1],"right":[14,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"0\",\"1\"],\"operation\":\"product\",\"right\":[\"14\",\"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-r03-factorization-0014","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [1, 1] 與 [15, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[15, 16, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [15, 16, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033014,"parameters":{"left":[1,1],"right":[15,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"1\",\"1\"],\"operation\":\"product\",\"right\":[\"15\",\"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-r03-factorization-0015","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [2, 1] 與 [16, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[32, 18, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [32, 18, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033015,"parameters":{"left":[2,1],"right":[16,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"16\",\"1\"],\"operation\":\"product\",\"right\":[\"2\",\"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-r03-factorization-0016","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [3, 1] 與 [17, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[51, 20, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [51, 20, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033016,"parameters":{"left":[3,1],"right":[17,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"17\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0017","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [4, 1] 與 [18, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[72, 22, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [72, 22, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033017,"parameters":{"left":[4,1],"right":[18,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"18\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0018","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [5, 1] 與 [19, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[95, 24, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [95, 24, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033018,"parameters":{"left":[5,1],"right":[19,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"19\",\"1\"],\"operation\":\"product\",\"right\":[\"5\",\"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-r03-factorization-0019","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [6, 1] 與 [20, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[120, 26, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [120, 26, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033019,"parameters":{"left":[6,1],"right":[20,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"20\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0020","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [7, 1] 與 [21, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[147, 28, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [147, 28, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033020,"parameters":{"left":[7,1],"right":[21,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"21\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0021","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [8, 1] 與 [22, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[176, 30, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [176, 30, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033021,"parameters":{"left":[8,1],"right":[22,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"22\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0022","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [9, 1] 與 [23, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[207, 32, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [207, 32, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033022,"parameters":{"left":[9,1],"right":[23,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"23\",\"1\"],\"operation\":\"product\",\"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-r03-factorization-0023","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [10, 1] 與 [24, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[240, 34, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [240, 34, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033023,"parameters":{"left":[10,1],"right":[24,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"10\",\"1\"],\"operation\":\"product\",\"right\":[\"24\",\"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-r03-factorization-0024","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [11, 1] 與 [25, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[275, 36, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [275, 36, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033024,"parameters":{"left":[11,1],"right":[25,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"11\",\"1\"],\"operation\":\"product\",\"right\":[\"25\",\"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-r03-factorization-0025","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的完整係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [12, 1] 與 [26, 1] 相乘後，規範係數向量為何？","answer":{"kind":"expression","value":"[312, 38, 1]","explanation_zh":"逐項相乘並按指數和合併係數，乘積為 [312, 38, 1]。比較的是完整係數，不是有限代入值。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的完整係數"]},"variant":{"family_id":"qf-factorization-product","seed":202610033025,"parameters":{"left":[12,1],"right":[26,1],"operation":"product","semantic_template":"qf-factorization-product/{\"left\":[\"12\",\"1\"],\"operation\":\"product\",\"right\":[\"26\",\"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-r03-factorization-0026","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [0] 與 [7, 1] 的乘積是否等於候選多項式 [0]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [0]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033026,"parameters":{"left":[0],"right":[7,1],"operation":"identity","proposed":[0],"semantic_template":"qf-factorization-identity/{\"left\":[\"7\",\"1\"],\"operation\":\"identity\",\"proposed\":[],\"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-r03-factorization-0027","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-11, 1] 與 [3, 1] 的乘積是否等於候選多項式 [-32, -8, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-33, -8, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033027,"parameters":{"left":[-11,1],"right":[3,1],"operation":"identity","proposed":[-32,-8,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"-11\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-32\",\"-8\",\"1\"],\"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-r03-factorization-0028","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-10, 1] 與 [4, 1] 的乘積是否等於候選多項式 [-40, -6, 2]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-40, -6, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033028,"parameters":{"left":[-10,1],"right":[4,1],"operation":"identity","proposed":[-40,-6,2],"semantic_template":"qf-factorization-identity/{\"left\":[\"-10\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-40\",\"-6\",\"2\"],\"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-r03-factorization-0029","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-9, 1] 與 [5, 1] 的乘積是否等於候選多項式 [-45, -4, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-45, -4, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033029,"parameters":{"left":[-9,1],"right":[5,1],"operation":"identity","proposed":[-45,-4,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"-9\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-45\",\"-4\",\"1\"],\"right\":[\"5\",\"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-r03-factorization-0030","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-8, 1] 與 [6, 1] 的乘積是否等於候選多項式 [-47, -2, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-48, -2, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033030,"parameters":{"left":[-8,1],"right":[6,1],"operation":"identity","proposed":[-47,-2,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"-8\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-47\",\"-2\",\"1\"],\"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-r03-factorization-0031","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-7, 1] 與 [7, 1] 的乘積是否等於候選多項式 [-49, 0, 2]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-49, 0, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033031,"parameters":{"left":[-7,1],"right":[7,1],"operation":"identity","proposed":[-49,0,2],"semantic_template":"qf-factorization-identity/{\"left\":[\"-7\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-49\",\"0\",\"2\"],\"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-r03-factorization-0032","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-6, 1] 與 [8, 1] 的乘積是否等於候選多項式 [-48, 2, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-48, 2, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033032,"parameters":{"left":[-6,1],"right":[8,1],"operation":"identity","proposed":[-48,2,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"-6\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-48\",\"2\",\"1\"],\"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-r03-factorization-0033","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-5, 1] 與 [9, 1] 的乘積是否等於候選多項式 [-44, 4, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-45, 4, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033033,"parameters":{"left":[-5,1],"right":[9,1],"operation":"identity","proposed":[-44,4,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"-5\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-44\",\"4\",\"1\"],\"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-r03-factorization-0034","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-4, 1] 與 [10, 1] 的乘積是否等於候選多項式 [-40, 6, 2]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-40, 6, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033034,"parameters":{"left":[-4,1],"right":[10,1],"operation":"identity","proposed":[-40,6,2],"semantic_template":"qf-factorization-identity/{\"left\":[\"-4\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-40\",\"6\",\"2\"],\"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-r03-factorization-0035","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-3, 1] 與 [11, 1] 的乘積是否等於候選多項式 [-33, 8, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-33, 8, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033035,"parameters":{"left":[-3,1],"right":[11,1],"operation":"identity","proposed":[-33,8,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"-3\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-33\",\"8\",\"1\"],\"right\":[\"11\",\"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-r03-factorization-0036","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-2, 1] 與 [12, 1] 的乘積是否等於候選多項式 [-23, 10, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-24, 10, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033036,"parameters":{"left":[-2,1],"right":[12,1],"operation":"identity","proposed":[-23,10,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"-2\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-23\",\"10\",\"1\"],\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r03-factorization-0037","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [-1, 1] 與 [13, 1] 的乘積是否等於候選多項式 [-13, 12, 2]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-13, 12, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033037,"parameters":{"left":[-1,1],"right":[13,1],"operation":"identity","proposed":[-13,12,2],"semantic_template":"qf-factorization-identity/{\"left\":[\"-1\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-13\",\"12\",\"2\"],\"right\":[\"13\",\"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-r03-factorization-0038","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [0, 1] 與 [14, 1] 的乘積是否等於候選多項式 [0, 14, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [0, 14, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033038,"parameters":{"left":[0,1],"right":[14,1],"operation":"identity","proposed":[0,14,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"0\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"0\",\"14\",\"1\"],\"right\":[\"14\",\"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-r03-factorization-0039","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [1, 1] 與 [15, 1] 的乘積是否等於候選多項式 [16, 16, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [15, 16, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033039,"parameters":{"left":[1,1],"right":[15,1],"operation":"identity","proposed":[16,16,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"1\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"16\",\"16\",\"1\"],\"right\":[\"15\",\"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-r03-factorization-0040","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [2, 1] 與 [16, 1] 的乘積是否等於候選多項式 [32, 18, 2]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [32, 18, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033040,"parameters":{"left":[2,1],"right":[16,1],"operation":"identity","proposed":[32,18,2],"semantic_template":"qf-factorization-identity/{\"left\":[\"16\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"32\",\"18\",\"2\"],\"right\":[\"2\",\"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-r03-factorization-0041","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [3, 1] 與 [17, 1] 的乘積是否等於候選多項式 [51, 20, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [51, 20, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033041,"parameters":{"left":[3,1],"right":[17,1],"operation":"identity","proposed":[51,20,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"17\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"51\",\"20\",\"1\"],\"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-r03-factorization-0042","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [4, 1] 與 [18, 1] 的乘積是否等於候選多項式 [73, 22, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [72, 22, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033042,"parameters":{"left":[4,1],"right":[18,1],"operation":"identity","proposed":[73,22,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"18\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"73\",\"22\",\"1\"],\"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-r03-factorization-0043","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [5, 1] 與 [19, 1] 的乘積是否等於候選多項式 [95, 24, 2]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [95, 24, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033043,"parameters":{"left":[5,1],"right":[19,1],"operation":"identity","proposed":[95,24,2],"semantic_template":"qf-factorization-identity/{\"left\":[\"19\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"95\",\"24\",\"2\"],\"right\":[\"5\",\"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-r03-factorization-0044","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [6, 1] 與 [20, 1] 的乘積是否等於候選多項式 [120, 26, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [120, 26, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033044,"parameters":{"left":[6,1],"right":[20,1],"operation":"identity","proposed":[120,26,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"20\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"120\",\"26\",\"1\"],\"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"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261003-r03-factorization-0046","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [8, 1] 與 [22, 1] 的乘積是否等於候選多項式 [176, 30, 2]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [176, 30, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033046,"parameters":{"left":[8,1],"right":[22,1],"operation":"identity","proposed":[176,30,2],"semantic_template":"qf-factorization-identity/{\"left\":[\"22\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"176\",\"30\",\"2\"],\"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-r03-factorization-0047","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [9, 1] 與 [23, 1] 的乘積是否等於候選多項式 [207, 32, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [207, 32, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033047,"parameters":{"left":[9,1],"right":[23,1],"operation":"identity","proposed":[207,32,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"23\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"207\",\"32\",\"1\"],\"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-r03-factorization-0048","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [10, 1] 與 [24, 1] 的乘積是否等於候選多項式 [241, 34, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [240, 34, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033048,"parameters":{"left":[10,1],"right":[24,1],"operation":"identity","proposed":[241,34,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"10\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"241\",\"34\",\"1\"],\"right\":[\"24\",\"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-r03-factorization-0049","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [11, 1] 與 [25, 1] 的乘積是否等於候選多項式 [275, 36, 2]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [275, 36, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033049,"parameters":{"left":[11,1],"right":[25,1],"operation":"identity","proposed":[275,36,2],"semantic_template":"qf-factorization-identity/{\"left\":[\"11\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"275\",\"36\",\"2\"],\"right\":[\"25\",\"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-r03-factorization-0050","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證因式分解係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。因子 [12, 1] 與 [26, 1] 的乘積是否等於候選多項式 [312, 38, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [312, 38, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證因式分解係數身份"]},"variant":{"family_id":"qf-factorization-identity","seed":202610033050,"parameters":{"left":[12,1],"right":[26,1],"operation":"identity","proposed":[312,38,1],"semantic_template":"qf-factorization-identity/{\"left\":[\"12\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"312\",\"38\",\"1\"],\"right\":[\"26\",\"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-r03-factorization-0051","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[-5, 1] 與 [2, -1] 的乘積中，x^0 的係數是多少？","answer":{"kind":"numeric","value":"-10","explanation_zh":"a_0=-10。把指數和為 0 的係數乘積相加；乘積完整係數為 [-10, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033051,"parameters":{"left":[-5,1],"right":[2,-1],"k":0,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"-5\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"2\",\"-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-r03-factorization-0052","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[-4, 1, 1] 與 [3, -1] 的乘積中，x^1 的係數是多少？","answer":{"kind":"numeric","value":"7","explanation_zh":"a_1=7。把指數和為 1 的係數乘積相加；乘積完整係數為 [-12, 7, 2, -1]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033052,"parameters":{"left":[-4,1,1],"right":[3,-1],"k":1,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":1,\"left\":[\"-4\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0053","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[-3, 1] 與 [4, -1] 的乘積中，x^2 的係數是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_2=-1。把指數和為 2 的係數乘積相加；乘積完整係數為 [-12, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033053,"parameters":{"left":[-3,1],"right":[4,-1],"k":2,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":2,\"left\":[\"-3\",\"1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0054","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[-2, 1, 1] 與 [5, -1] 的乘積中，x^3 的係數是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_3=-1。把指數和為 3 的係數乘積相加；乘積完整係數為 [-10, 7, 4, -1]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033054,"parameters":{"left":[-2,1,1],"right":[5,-1],"k":3,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":3,\"left\":[\"-2\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"5\",\"-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-r03-factorization-0055","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[-1, 1] 與 [6, -1] 的乘積中，x^4 的係數是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。把指數和為 4 的係數乘積相加；乘積完整係數為 [-6, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033055,"parameters":{"left":[-1,1],"right":[6,-1],"k":4,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":4,\"left\":[\"-1\",\"1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0056","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[0, 1, 1] 與 [7, -1] 的乘積中，x^5 的係數是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_5=0。把指數和為 5 的係數乘積相加；乘積完整係數為 [0, 7, 6, -1]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033056,"parameters":{"left":[0,1,1],"right":[7,-1],"k":5,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":5,\"left\":[\"0\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0057","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[1, 1] 與 [8, -1] 的乘積中，x^0 的係數是多少？","answer":{"kind":"numeric","value":"8","explanation_zh":"a_0=8。把指數和為 0 的係數乘積相加；乘積完整係數為 [8, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033057,"parameters":{"left":[1,1],"right":[8,-1],"k":0,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"1\",\"1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0058","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[2, 1, 1] 與 [9, -1] 的乘積中，x^1 的係數是多少？","answer":{"kind":"numeric","value":"7","explanation_zh":"a_1=7。把指數和為 1 的係數乘積相加；乘積完整係數為 [18, 7, 8, -1]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033058,"parameters":{"left":[2,1,1],"right":[9,-1],"k":1,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":1,\"left\":[\"2\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0059","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[3, 1] 與 [10, -1] 的乘積中，x^2 的係數是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_2=-1。把指數和為 2 的係數乘積相加；乘積完整係數為 [30, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033059,"parameters":{"left":[3,1],"right":[10,-1],"k":2,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":2,\"left\":[\"10\",\"-1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0060","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[4, 1, 1] 與 [11, -1] 的乘積中，x^3 的係數是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_3=-1。把指數和為 3 的係數乘積相加；乘積完整係數為 [44, 7, 10, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033060,"parameters":{"left":[4,1,1],"right":[11,-1],"k":3,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":3,\"left\":[\"11\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"4\",\"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-r03-factorization-0061","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[5, 1] 與 [12, -1] 的乘積中，x^4 的係數是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。把指數和為 4 的係數乘積相加；乘積完整係數為 [60, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033061,"parameters":{"left":[5,1],"right":[12,-1],"k":4,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":4,\"left\":[\"12\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"5\",\"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-r03-factorization-0062","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[6, 1, 1] 與 [13, -1] 的乘積中，x^5 的係數是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_5=0。把指數和為 5 的係數乘積相加；乘積完整係數為 [78, 7, 12, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033062,"parameters":{"left":[6,1,1],"right":[13,-1],"k":5,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":5,\"left\":[\"13\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"6\",\"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-r03-factorization-0063","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[7, 1] 與 [14, -1] 的乘積中，x^0 的係數是多少？","answer":{"kind":"numeric","value":"98","explanation_zh":"a_0=98。把指數和為 0 的係數乘積相加；乘積完整係數為 [98, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033063,"parameters":{"left":[7,1],"right":[14,-1],"k":0,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"14\",\"-1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0064","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[8, 1, 1] 與 [15, -1] 的乘積中，x^1 的係數是多少？","answer":{"kind":"numeric","value":"7","explanation_zh":"a_1=7。把指數和為 1 的係數乘積相加；乘積完整係數為 [120, 7, 14, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033064,"parameters":{"left":[8,1,1],"right":[15,-1],"k":1,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":1,\"left\":[\"15\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"8\",\"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-r03-factorization-0065","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[9, 1] 與 [16, -1] 的乘積中，x^2 的係數是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_2=-1。把指數和為 2 的係數乘積相加；乘積完整係數為 [144, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033065,"parameters":{"left":[9,1],"right":[16,-1],"k":2,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":2,\"left\":[\"16\",\"-1\"],\"operation\":\"product_coefficient\",\"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-r03-factorization-0066","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[10, 1, 1] 與 [17, -1] 的乘積中，x^3 的係數是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_3=-1。把指數和為 3 的係數乘積相加；乘積完整係數為 [170, 7, 16, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033066,"parameters":{"left":[10,1,1],"right":[17,-1],"k":3,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":3,\"left\":[\"10\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"17\",\"-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-r03-factorization-0067","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[11, 1] 與 [18, -1] 的乘積中，x^4 的係數是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。把指數和為 4 的係數乘積相加；乘積完整係數為 [198, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033067,"parameters":{"left":[11,1],"right":[18,-1],"k":4,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":4,\"left\":[\"11\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"18\",\"-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-r03-factorization-0068","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[12, 1, 1] 與 [19, -1] 的乘積中，x^5 的係數是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_5=0。把指數和為 5 的係數乘積相加；乘積完整係數為 [228, 7, 18, -1]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033068,"parameters":{"left":[12,1,1],"right":[19,-1],"k":5,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":5,\"left\":[\"12\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"19\",\"-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-r03-factorization-0069","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[13, 1] 與 [20, -1] 的乘積中，x^0 的係數是多少？","answer":{"kind":"numeric","value":"260","explanation_zh":"a_0=260。把指數和為 0 的係數乘積相加；乘積完整係數為 [260, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033069,"parameters":{"left":[13,1],"right":[20,-1],"k":0,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"13\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"20\",\"-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-r03-factorization-0070","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[14, 1, 1] 與 [21, -1] 的乘積中，x^1 的係數是多少？","answer":{"kind":"numeric","value":"7","explanation_zh":"a_1=7。把指數和為 1 的係數乘積相加；乘積完整係數為 [294, 7, 20, -1]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033070,"parameters":{"left":[14,1,1],"right":[21,-1],"k":1,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":1,\"left\":[\"14\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"21\",\"-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-r03-factorization-0071","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[15, 1] 與 [22, -1] 的乘積中，x^2 的係數是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_2=-1。把指數和為 2 的係數乘積相加；乘積完整係數為 [330, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033071,"parameters":{"left":[15,1],"right":[22,-1],"k":2,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":2,\"left\":[\"15\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"22\",\"-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-r03-factorization-0072","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[16, 1, 1] 與 [23, -1] 的乘積中，x^3 的係數是多少？","answer":{"kind":"numeric","value":"-1","explanation_zh":"a_3=-1。把指數和為 3 的係數乘積相加；乘積完整係數為 [368, 7, 22, -1]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033072,"parameters":{"left":[16,1,1],"right":[23,-1],"k":3,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":3,\"left\":[\"16\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"23\",\"-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-r03-factorization-0073","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[17, 1] 與 [24, -1] 的乘積中，x^4 的係數是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_4=0。把指數和為 4 的係數乘積相加；乘積完整係數為 [408, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033073,"parameters":{"left":[17,1],"right":[24,-1],"k":4,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":4,\"left\":[\"17\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"24\",\"-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-r03-factorization-0074","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[18, 1, 1] 與 [25, -1] 的乘積中，x^5 的係數是多少？","answer":{"kind":"numeric","value":"0","explanation_zh":"a_5=0。把指數和為 5 的係數乘積相加；乘積完整係數為 [450, 7, 24, -1]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033074,"parameters":{"left":[18,1,1],"right":[25,-1],"k":5,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":5,\"left\":[\"18\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"25\",\"-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-r03-factorization-0075","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"因子乘積的指定係數","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。[19, 1] 與 [26, -1] 的乘積中，x^0 的係數是多少？","answer":{"kind":"numeric","value":"494","explanation_zh":"a_0=494。把指數和為 0 的係數乘積相加；乘積完整係數為 [494, 7, -1]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["因子乘積的指定係數"]},"variant":{"family_id":"qf-factorization-product-coefficient","seed":202610033075,"parameters":{"left":[19,1],"right":[26,-1],"k":0,"operation":"product_coefficient","semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"19\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"26\",\"-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-r03-factorization-0076","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(0)² 是否等於候選係數向量 [0, 0, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [0, 0, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033076,"parameters":{"d":0,"proposed":[0,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":0,\"operation\":\"difference_squares\",\"proposed\":[\"0\",\"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-r03-factorization-0077","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(1)² 是否等於候選係數向量 [0, 0, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-1, 0, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033077,"parameters":{"d":1,"proposed":[0,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":1,\"operation\":\"difference_squares\",\"proposed\":[\"0\",\"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-r03-factorization-0078","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(2)² 是否等於候選係數向量 [-4, 0, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-4, 0, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033078,"parameters":{"d":2,"proposed":[-4,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":2,\"operation\":\"difference_squares\",\"proposed\":[\"-4\",\"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-r03-factorization-0079","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(3)² 是否等於候選係數向量 [-8, 0, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-9, 0, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033079,"parameters":{"d":3,"proposed":[-8,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":3,\"operation\":\"difference_squares\",\"proposed\":[\"-8\",\"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-r03-factorization-0080","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(4)² 是否等於候選係數向量 [-16, 0, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-16, 0, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033080,"parameters":{"d":4,"proposed":[-16,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":4,\"operation\":\"difference_squares\",\"proposed\":[\"-16\",\"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-r03-factorization-0081","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(5)² 是否等於候選係數向量 [-24, 0, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-25, 0, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033081,"parameters":{"d":5,"proposed":[-24,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":5,\"operation\":\"difference_squares\",\"proposed\":[\"-24\",\"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-r03-factorization-0082","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(6)² 是否等於候選係數向量 [-36, 0, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-36, 0, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033082,"parameters":{"d":6,"proposed":[-36,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":6,\"operation\":\"difference_squares\",\"proposed\":[\"-36\",\"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-r03-factorization-0083","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(7)² 是否等於候選係數向量 [-48, 0, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-49, 0, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033083,"parameters":{"d":7,"proposed":[-48,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":7,\"operation\":\"difference_squares\",\"proposed\":[\"-48\",\"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-r03-factorization-0084","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(8)² 是否等於候選係數向量 [-64, 0, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-64, 0, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033084,"parameters":{"d":8,"proposed":[-64,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":8,\"operation\":\"difference_squares\",\"proposed\":[\"-64\",\"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-r03-factorization-0085","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(9)² 是否等於候選係數向量 [-80, 0, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-81, 0, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033085,"parameters":{"d":9,"proposed":[-80,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":9,\"operation\":\"difference_squares\",\"proposed\":[\"-80\",\"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"}}
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{"schema_version":"ocme-question-v0.1","id":"q-20261003-r03-factorization-0096","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(20)² 是否等於候選係數向量 [-400, 0, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-400, 0, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033096,"parameters":{"d":20,"proposed":[-400,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":20,\"operation\":\"difference_squares\",\"proposed\":[\"-400\",\"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-r03-factorization-0097","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(21)² 是否等於候選係數向量 [-440, 0, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-441, 0, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033097,"parameters":{"d":21,"proposed":[-440,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":21,\"operation\":\"difference_squares\",\"proposed\":[\"-440\",\"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-r03-factorization-0098","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(22)² 是否等於候選係數向量 [-484, 0, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-484, 0, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033098,"parameters":{"d":22,"proposed":[-484,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":22,\"operation\":\"difference_squares\",\"proposed\":[\"-484\",\"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-r03-factorization-0099","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(23)² 是否等於候選係數向量 [-528, 0, 1]？","answer":{"kind":"short_text","value":"否","explanation_zh":"正確展開的係數向量為 [-529, 0, 1]，與候選逐項比較後答案是否。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033099,"parameters":{"d":23,"proposed":[-528,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":23,\"operation\":\"difference_squares\",\"proposed\":[\"-528\",\"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-r03-factorization-0100","batch_id":"batch-20261003-r03-factorization-foundations","language":"zh-Hant","source_mko_ids":["mko-factorization"],"title_zh":"驗證平方差係數身份","stem_zh":"在 Q[x] 中，係數向量由低次至高次排列。平方差 x²−(24)² 是否等於候選係數向量 [-576, 0, 1]？","answer":{"kind":"short_text","value":"是","explanation_zh":"正確展開的係數向量為 [-576, 0, 1]，與候選逐項比較後答案是是。這是係數恆等驗證，並非數值抽樣。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["驗證平方差係數身份"]},"variant":{"family_id":"qf-factorization-difference-squares","seed":202610033100,"parameters":{"d":24,"proposed":[-576,0,1],"operation":"difference_squares","semantic_template":"qf-factorization-difference-squares/{\"d\":24,\"operation\":\"difference_squares\",\"proposed\":[\"-576\",\"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"}}