{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0001","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[0]、g=[0]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[0]","explanation_zh":"和的升冪係數=[0]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101001,"parameters":{"f":[0],"g":[0],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[0],[0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0002","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[2]、g=[-3]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-1]","explanation_zh":"和的升冪係數=[-1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101002,"parameters":{"f":[2],"g":[-3],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-3],[2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0003","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[3, -2, 1]、g=[-3, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[0]","explanation_zh":"和的升冪係數=[0]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101003,"parameters":{"f":[3,-2,1],"g":[-3,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-3,2,-1],[3,-2,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0004","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-9, 4, -1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-8, 2, 1, 1]","explanation_zh":"和的升冪係數=[-8, 2, 1, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101004,"parameters":{"f":[-9,4,-1,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-9,4,-1,2],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0005","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-8, 5, 0, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-7, 3, 2]","explanation_zh":"和的升冪係數=[-7, 3, 2]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101005,"parameters":{"f":[-8,5,0,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-8,5,0,1],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0006","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-7, 6, 1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-6, 4, 3, 1]","explanation_zh":"和的升冪係數=[-6, 4, 3, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101006,"parameters":{"f":[-7,6,1,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-7,6,1,2],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0007","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-6, 7, -1, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-5, 5, 1]","explanation_zh":"和的升冪係數=[-5, 5, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101007,"parameters":{"f":[-6,7,-1,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-6,7,-1,1],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0008","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-5, 8, 0, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-4, 6, 2, 1]","explanation_zh":"和的升冪係數=[-4, 6, 2, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101008,"parameters":{"f":[-5,8,0,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-5,8,0,2],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0009","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-4, 9, 1, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-3, 7, 3]","explanation_zh":"和的升冪係數=[-3, 7, 3]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101009,"parameters":{"f":[-4,9,1,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-4,9,1,1],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0010","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-3, 10, -1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-2, 8, 1, 1]","explanation_zh":"和的升冪係數=[-2, 8, 1, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101010,"parameters":{"f":[-3,10,-1,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-3,10,-1,2],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0011","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-2, 11, 0, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[-1, 9, 2]","explanation_zh":"和的升冪係數=[-1, 9, 2]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101011,"parameters":{"f":[-2,11,0,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-2,11,0,1],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0012","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[-1, 12, 1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[0, 10, 3, 1]","explanation_zh":"和的升冪係數=[0, 10, 3, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101012,"parameters":{"f":[-1,12,1,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[-1,12,1,2],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0013","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[0, 13, -1, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[1, 11, 1]","explanation_zh":"和的升冪係數=[1, 11, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101013,"parameters":{"f":[0,13,-1,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[0,13,-1,1],[1,-2,2,-1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0014","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[1, 14, 0, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[2, 12, 2, 1]","explanation_zh":"和的升冪係數=[2, 12, 2, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101014,"parameters":{"f":[1,14,0,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[1,14,0,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0015","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[2, 15, 1, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[3, 13, 3]","explanation_zh":"和的升冪係數=[3, 13, 3]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101015,"parameters":{"f":[2,15,1,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[2,15,1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0016","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[3, 16, -1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[4, 14, 1, 1]","explanation_zh":"和的升冪係數=[4, 14, 1, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101016,"parameters":{"f":[3,16,-1,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[3,16,-1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0017","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[4, 17, 0, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[5, 15, 2]","explanation_zh":"和的升冪係數=[5, 15, 2]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101017,"parameters":{"f":[4,17,0,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[4,17,0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0018","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[5, 18, 1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[6, 16, 3, 1]","explanation_zh":"和的升冪係數=[6, 16, 3, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101018,"parameters":{"f":[5,18,1,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[5,18,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0019","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[6, 19, -1, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[7, 17, 1]","explanation_zh":"和的升冪係數=[7, 17, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101019,"parameters":{"f":[6,19,-1,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[6,19,-1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0020","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[7, 20, 0, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[8, 18, 2, 1]","explanation_zh":"和的升冪係數=[8, 18, 2, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101020,"parameters":{"f":[7,20,0,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[7,20,0,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0021","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[8, 21, 1, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[9, 19, 3]","explanation_zh":"和的升冪係數=[9, 19, 3]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101021,"parameters":{"f":[8,21,1,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[8,21,1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0022","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[9, 22, -1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[10, 20, 1, 1]","explanation_zh":"和的升冪係數=[10, 20, 1, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101022,"parameters":{"f":[9,22,-1,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[9,22,-1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0023","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[10, 23, 0, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[11, 21, 2]","explanation_zh":"和的升冪係數=[11, 21, 2]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101023,"parameters":{"f":[10,23,0,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[10,23,0,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0024","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[11, 24, 1, 2]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[12, 22, 3, 1]","explanation_zh":"和的升冪係數=[12, 22, 3, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101024,"parameters":{"f":[11,24,1,2],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[11,24,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0025","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式和的完整係數","stem_zh":"升冪係數 f=[12, 25, -1, 1]、g=[1, -2, 2, -1]。求 f+g 的完整升冪係數列表。","answer":{"kind":"expression","value":"[13, 23, 1]","explanation_zh":"和的升冪係數=[13, 23, 1]。逐次冪合併係數並去尾端零；零多項式保留[0]，不由單點值判定恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式和的完整係數"]},"variant":{"family_id":"qf-density10-poly-add-coefficients","seed":202610101025,"parameters":{"f":[12,25,-1,1],"g":[1,-2,2,-1],"representation":"ascending_powers","operation":"density10_poly_add_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[12,25,-1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0026","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[0]、g=[1, -1]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[0]","explanation_zh":"積的升冪係數=[0]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101026,"parameters":{"f":[0],"g":[1,-1],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[\"zero_product\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0027","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[2]、g=[-3]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-6]","explanation_zh":"積的升冪係數=[-6]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101027,"parameters":{"f":[2],"g":[-3],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-3],[2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0028","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[1, 1]、g=[1, -1]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[1, 0, -1]","explanation_zh":"積的升冪係數=[1, 0, -1]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101028,"parameters":{"f":[1,1],"g":[1,-1],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[1,-1],[1,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0029","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-9, 4, -3, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-18, 17, -19, -9, 4, -5, 2]","explanation_zh":"積的升冪係數=[-18, 17, -19, -9, 4, -5, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101029,"parameters":{"f":[-9,4,-3,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-9,4,-3,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0030","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-8, 5, -4, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-16, 18, -21, -5, 5, -7, 2]","explanation_zh":"積的升冪係數=[-16, 18, -21, -5, 5, -7, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101030,"parameters":{"f":[-8,5,-4,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-8,5,-4,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0031","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-7, 6, -5, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-14, 19, -23, -1, 6, -9, 2]","explanation_zh":"積的升冪係數=[-14, 19, -23, -1, 6, -9, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101031,"parameters":{"f":[-7,6,-5,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-7,6,-5,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0032","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-6, 7, -6, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-12, 20, -25, 3, 7, -11, 2]","explanation_zh":"積的升冪係數=[-12, 20, -25, 3, 7, -11, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101032,"parameters":{"f":[-6,7,-6,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-6,7,-6,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0033","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-5, 8, -7, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-10, 21, -27, 7, 8, -13, 2]","explanation_zh":"積的升冪係數=[-10, 21, -27, 7, 8, -13, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101033,"parameters":{"f":[-5,8,-7,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-5,8,-7,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0034","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-4, 9, -8, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-8, 22, -29, 11, 9, -15, 2]","explanation_zh":"積的升冪係數=[-8, 22, -29, 11, 9, -15, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101034,"parameters":{"f":[-4,9,-8,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-4,9,-8,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0035","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-3, 10, -9, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-6, 23, -31, 15, 10, -17, 2]","explanation_zh":"積的升冪係數=[-6, 23, -31, 15, 10, -17, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101035,"parameters":{"f":[-3,10,-9,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-3,10,-9,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0036","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-2, 11, -10, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-4, 24, -33, 19, 11, -19, 2]","explanation_zh":"積的升冪係數=[-4, 24, -33, 19, 11, -19, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101036,"parameters":{"f":[-2,11,-10,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-2,11,-10,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0037","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[-1, 12, -11, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[-2, 25, -35, 23, 12, -21, 2]","explanation_zh":"積的升冪係數=[-2, 25, -35, 23, 12, -21, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101037,"parameters":{"f":[-1,12,-11,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[-1,12,-11,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0038","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[0, 13, -12, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[0, 26, -37, 27, 13, -23, 2]","explanation_zh":"積的升冪係數=[0, 26, -37, 27, 13, -23, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101038,"parameters":{"f":[0,13,-12,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[0,13,-12,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0039","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[1, 14, -13, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[2, 27, -39, 31, 14, -25, 2]","explanation_zh":"積的升冪係數=[2, 27, -39, 31, 14, -25, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101039,"parameters":{"f":[1,14,-13,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[1,14,-13,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0040","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[2, 15, -14, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[4, 28, -41, 35, 15, -27, 2]","explanation_zh":"積的升冪係數=[4, 28, -41, 35, 15, -27, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101040,"parameters":{"f":[2,15,-14,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[2,15,-14,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0041","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[3, 16, -15, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[6, 29, -43, 39, 16, -29, 2]","explanation_zh":"積的升冪係數=[6, 29, -43, 39, 16, -29, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101041,"parameters":{"f":[3,16,-15,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[3,16,-15,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0042","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[4, 17, -16, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[8, 30, -45, 43, 17, -31, 2]","explanation_zh":"積的升冪係數=[8, 30, -45, 43, 17, -31, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101042,"parameters":{"f":[4,17,-16,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[4,17,-16,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0043","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[5, 18, -17, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[10, 31, -47, 47, 18, -33, 2]","explanation_zh":"積的升冪係數=[10, 31, -47, 47, 18, -33, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101043,"parameters":{"f":[5,18,-17,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[5,18,-17,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0044","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[6, 19, -18, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[12, 32, -49, 51, 19, -35, 2]","explanation_zh":"積的升冪係數=[12, 32, -49, 51, 19, -35, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101044,"parameters":{"f":[6,19,-18,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[6,19,-18,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0045","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[7, 20, -19, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[14, 33, -51, 55, 20, -37, 2]","explanation_zh":"積的升冪係數=[14, 33, -51, 55, 20, -37, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101045,"parameters":{"f":[7,20,-19,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[7,20,-19,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0046","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[8, 21, -20, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[16, 34, -53, 59, 21, -39, 2]","explanation_zh":"積的升冪係數=[16, 34, -53, 59, 21, -39, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101046,"parameters":{"f":[8,21,-20,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[8,21,-20,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0047","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[9, 22, -21, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[18, 35, -55, 63, 22, -41, 2]","explanation_zh":"積的升冪係數=[18, 35, -55, 63, 22, -41, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101047,"parameters":{"f":[9,22,-21,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[9,22,-21,1]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0048","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[10, 23, -22, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[20, 36, -57, 67, 23, -43, 2]","explanation_zh":"積的升冪係數=[20, 36, -57, 67, 23, -43, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101048,"parameters":{"f":[10,23,-22,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[10,23,-22,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0049","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[11, 24, -23, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[22, 37, -59, 71, 24, -45, 2]","explanation_zh":"積的升冪係數=[22, 37, -59, 71, 24, -45, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101049,"parameters":{"f":[11,24,-23,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[11,24,-23,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0050","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"多項式積的完整係數","stem_zh":"升冪係數 f=[12, 25, -24, 1]、g=[2, -1, 1, 2]。求 fg 的完整升冪係數列表，最高可至6次，不截斷。","answer":{"kind":"expression","value":"[24, 38, -61, 75, 25, -47, 2]","explanation_zh":"積的升冪係數=[24, 38, -61, 75, 25, -47, 2]。以多項式Horner遞推乘入另一因子；保留所有次冪再去尾端零，零因子所得為[0]。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["多項式積的完整係數"]},"variant":{"family_id":"qf-density10-poly-product-coefficients","seed":202610101050,"parameters":{"f":[12,25,-24,1],"g":[2,-1,1,2],"representation":"ascending_powers","operation":"density10_poly_product_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-product-coefficients/[[12,25,-24,1],[2,-1,1,2]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0051","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[0]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(0, 0)","explanation_zh":"有序雙點值=(0, 0)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101051,"parameters":{"f":[0],"a":-2,"b":1,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0052","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-3]，a=-2、b=2。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-3, -3)","explanation_zh":"有序雙點值=(-3, -3)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101052,"parameters":{"f":[-3],"a":-2,"b":2,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-3]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0053","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-10, 2, -1, 2]，a=-2、b=3。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-34, 41)","explanation_zh":"有序雙點值=(-34, 41)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101053,"parameters":{"f":[-10,2,-1,2],"a":-2,"b":3,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-10,2,-1,2],-2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0054","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-9, 2, -1, 0]，a=-2、b=4。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-17, -17)","explanation_zh":"有序雙點值=(-17, -17)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101054,"parameters":{"f":[-9,2,-1,0],"a":-2,"b":4,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-9,2,-1],-2,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0055","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-8, 2, -1, 1]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-24, -6)","explanation_zh":"有序雙點值=(-24, -6)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101055,"parameters":{"f":[-8,2,-1,1],"a":-2,"b":1,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-8,2,-1,1],-2,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0056","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-7, 2, -1, 2]，a=-2、b=2。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-31, 9)","explanation_zh":"有序雙點值=(-31, 9)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101056,"parameters":{"f":[-7,2,-1,2],"a":-2,"b":2,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-7,2,-1,2],-2,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0057","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-6, 2, -1, 0]，a=-2、b=3。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-14, -9)","explanation_zh":"有序雙點值=(-14, -9)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101057,"parameters":{"f":[-6,2,-1,0],"a":-2,"b":3,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-6,2,-1],-2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0058","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-5, 2, -1, 1]，a=-2、b=4。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-21, 51)","explanation_zh":"有序雙點值=(-21, 51)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101058,"parameters":{"f":[-5,2,-1,1],"a":-2,"b":4,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-5,2,-1,1],-2,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0059","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-4, 2, -1, 2]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-28, -1)","explanation_zh":"有序雙點值=(-28, -1)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101059,"parameters":{"f":[-4,2,-1,2],"a":-2,"b":1,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-4,2,-1,2],-2,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0060","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-3, 2, -1, 0]，a=-2、b=2。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-11, -3)","explanation_zh":"有序雙點值=(-11, -3)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101060,"parameters":{"f":[-3,2,-1,0],"a":-2,"b":2,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-3,2,-1],-2,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0061","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-2, 2, -1, 1]，a=-2、b=3。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-18, 22)","explanation_zh":"有序雙點值=(-18, 22)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101061,"parameters":{"f":[-2,2,-1,1],"a":-2,"b":3,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-2,2,-1,1],-2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0062","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[-1, 2, -1, 2]，a=-2、b=4。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-25, 119)","explanation_zh":"有序雙點值=(-25, 119)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101062,"parameters":{"f":[-1,2,-1,2],"a":-2,"b":4,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[-1,2,-1,2],-2,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0063","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[0, 2, -1, 0]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-8, 1)","explanation_zh":"有序雙點值=(-8, 1)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101063,"parameters":{"f":[0,2,-1,0],"a":-2,"b":1,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[0,2,-1],-2,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0064","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[1, 2, -1, 1]，a=-2、b=2。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-15, 9)","explanation_zh":"有序雙點值=(-15, 9)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101064,"parameters":{"f":[1,2,-1,1],"a":-2,"b":2,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[1,2,-1,1],-2,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0065","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[2, 2, -1, 2]，a=-2、b=3。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-22, 53)","explanation_zh":"有序雙點值=(-22, 53)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101065,"parameters":{"f":[2,2,-1,2],"a":-2,"b":3,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[2,2,-1,2],-2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0066","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[3, 2, -1, 0]，a=-2、b=4。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-5, -5)","explanation_zh":"有序雙點值=(-5, -5)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101066,"parameters":{"f":[3,2,-1,0],"a":-2,"b":4,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[3,2,-1],-2,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0067","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[4, 2, -1, 1]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-12, 6)","explanation_zh":"有序雙點值=(-12, 6)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101067,"parameters":{"f":[4,2,-1,1],"a":-2,"b":1,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[4,2,-1,1],-2,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0068","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[5, 2, -1, 2]，a=-2、b=2。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-19, 21)","explanation_zh":"有序雙點值=(-19, 21)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101068,"parameters":{"f":[5,2,-1,2],"a":-2,"b":2,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[5,2,-1,2],-2,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0069","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[6, 2, -1, 0]，a=-2、b=3。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-2, 3)","explanation_zh":"有序雙點值=(-2, 3)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101069,"parameters":{"f":[6,2,-1,0],"a":-2,"b":3,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[6,2,-1],-2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0070","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[7, 2, -1, 1]，a=-2、b=4。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-9, 63)","explanation_zh":"有序雙點值=(-9, 63)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101070,"parameters":{"f":[7,2,-1,1],"a":-2,"b":4,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[7,2,-1,1],-2,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0071","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[8, 2, -1, 2]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-16, 11)","explanation_zh":"有序雙點值=(-16, 11)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101071,"parameters":{"f":[8,2,-1,2],"a":-2,"b":1,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[8,2,-1,2],-2,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0072","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[9, 2, -1, 0]，a=-2、b=2。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(1, 9)","explanation_zh":"有序雙點值=(1, 9)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101072,"parameters":{"f":[9,2,-1,0],"a":-2,"b":2,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[9,2,-1],-2,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0073","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[10, 2, -1, 1]，a=-2、b=3。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-6, 34)","explanation_zh":"有序雙點值=(-6, 34)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101073,"parameters":{"f":[10,2,-1,1],"a":-2,"b":3,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[10,2,-1,1],-2,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0074","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[11, 2, -1, 2]，a=-2、b=4。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(-13, 131)","explanation_zh":"有序雙點值=(-13, 131)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101074,"parameters":{"f":[11,2,-1,2],"a":-2,"b":4,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[11,2,-1,2],-2,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0075","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"同一多項式的有序雙點值","stem_zh":"升冪係數 f=[12, 2, -1, 0]，a=-2、b=1。依序求 (f(a), f(b)) 的整數值。","answer":{"kind":"expression","value":"(4, 13)","explanation_zh":"有序雙點值=(4, 13)。分別用Horner代入兩點，保持a、b次序；兩點結果相同不代表多項式恆等。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["同一多項式的有序雙點值"]},"variant":{"family_id":"qf-density10-poly-two-point-values","seed":202610101075,"parameters":{"f":[12,2,-1,0],"a":-2,"b":1,"representation":"ascending_powers","operation":"density10_poly_two_point_values","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-two-point-values/[[12,2,-1],-2,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0076","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[0]，c=-2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[0]","explanation_zh":"平移後升冪係數=[0]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101076,"parameters":{"f":[0],"c":-2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[0]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0077","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-4]，c=-1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-4]","explanation_zh":"平移後升冪係數=[-4]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101077,"parameters":{"f":[-4],"c":-1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-4]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0078","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-10, 3, -2, 1]，c=0。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-10, 3, -2, 1]","explanation_zh":"平移後升冪係數=[-10, 3, -2, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101078,"parameters":{"f":[-10,3,-2,1],"c":0,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-10,3,-2,1],0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0079","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-9, 4, -2, 1]，c=1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-6, 3, 1, 1]","explanation_zh":"平移後升冪係數=[-6, 3, 1, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101079,"parameters":{"f":[-9,4,-2,1],"c":1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-9,4,-2,1],1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0080","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-8, 5, -2, 1]，c=2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[2, 9, 4, 1]","explanation_zh":"平移後升冪係數=[2, 9, 4, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101080,"parameters":{"f":[-8,5,-2,1],"c":2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-8,5,-2,1],2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0081","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-7, 6, -2, 1]，c=-2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-35, 26, -8, 1]","explanation_zh":"平移後升冪係數=[-35, 26, -8, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101081,"parameters":{"f":[-7,6,-2,1],"c":-2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-7,6,-2,1],-2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0082","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-6, 7, -2, 1]，c=-1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-16, 14, -5, 1]","explanation_zh":"平移後升冪係數=[-16, 14, -5, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101082,"parameters":{"f":[-6,7,-2,1],"c":-1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-6,7,-2,1],-1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0083","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-5, 8, -2, 1]，c=0。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-5, 8, -2, 1]","explanation_zh":"平移後升冪係數=[-5, 8, -2, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101083,"parameters":{"f":[-5,8,-2,1],"c":0,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-5,8,-2,1],0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0084","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-4, 9, -2, 1]，c=1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[4, 8, 1, 1]","explanation_zh":"平移後升冪係數=[4, 8, 1, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101084,"parameters":{"f":[-4,9,-2,1],"c":1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-4,9,-2,1],1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0085","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-3, 10, -2, 1]，c=2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[17, 14, 4, 1]","explanation_zh":"平移後升冪係數=[17, 14, 4, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101085,"parameters":{"f":[-3,10,-2,1],"c":2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-3,10,-2,1],2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0086","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-2, 11, -2, 1]，c=-2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-40, 31, -8, 1]","explanation_zh":"平移後升冪係數=[-40, 31, -8, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101086,"parameters":{"f":[-2,11,-2,1],"c":-2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-2,11,-2,1],-2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0087","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[-1, 12, -2, 1]，c=-1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-16, 19, -5, 1]","explanation_zh":"平移後升冪係數=[-16, 19, -5, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101087,"parameters":{"f":[-1,12,-2,1],"c":-1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[-1,12,-2,1],-1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0088","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[0, 13, -2, 1]，c=0。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[0, 13, -2, 1]","explanation_zh":"平移後升冪係數=[0, 13, -2, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101088,"parameters":{"f":[0,13,-2,1],"c":0,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[0,13,-2,1],0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0089","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[1, 14, -2, 1]，c=1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[14, 13, 1, 1]","explanation_zh":"平移後升冪係數=[14, 13, 1, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101089,"parameters":{"f":[1,14,-2,1],"c":1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[1,14,-2,1],1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0090","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[2, 15, -2, 1]，c=2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[32, 19, 4, 1]","explanation_zh":"平移後升冪係數=[32, 19, 4, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101090,"parameters":{"f":[2,15,-2,1],"c":2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[2,15,-2,1],2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0091","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[3, 16, -2, 1]，c=-2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-45, 36, -8, 1]","explanation_zh":"平移後升冪係數=[-45, 36, -8, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101091,"parameters":{"f":[3,16,-2,1],"c":-2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[3,16,-2,1],-2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0092","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[4, 17, -2, 1]，c=-1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-16, 24, -5, 1]","explanation_zh":"平移後升冪係數=[-16, 24, -5, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101092,"parameters":{"f":[4,17,-2,1],"c":-1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[4,17,-2,1],-1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0093","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[5, 18, -2, 1]，c=0。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[5, 18, -2, 1]","explanation_zh":"平移後升冪係數=[5, 18, -2, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101093,"parameters":{"f":[5,18,-2,1],"c":0,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[5,18,-2,1],0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0094","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[6, 19, -2, 1]，c=1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[24, 18, 1, 1]","explanation_zh":"平移後升冪係數=[24, 18, 1, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101094,"parameters":{"f":[6,19,-2,1],"c":1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[6,19,-2,1],1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0095","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[7, 20, -2, 1]，c=2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[47, 24, 4, 1]","explanation_zh":"平移後升冪係數=[47, 24, 4, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101095,"parameters":{"f":[7,20,-2,1],"c":2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[7,20,-2,1],2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0096","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[8, 21, -2, 1]，c=-2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-50, 41, -8, 1]","explanation_zh":"平移後升冪係數=[-50, 41, -8, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101096,"parameters":{"f":[8,21,-2,1],"c":-2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[8,21,-2,1],-2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0097","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[9, 22, -2, 1]，c=-1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[-16, 29, -5, 1]","explanation_zh":"平移後升冪係數=[-16, 29, -5, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101097,"parameters":{"f":[9,22,-2,1],"c":-1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[9,22,-2,1],-1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0098","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[10, 23, -2, 1]，c=0。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[10, 23, -2, 1]","explanation_zh":"平移後升冪係數=[10, 23, -2, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101098,"parameters":{"f":[10,23,-2,1],"c":0,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[10,23,-2,1],0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0099","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[11, 24, -2, 1]，c=1。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[34, 23, 1, 1]","explanation_zh":"平移後升冪係數=[34, 23, 1, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101099,"parameters":{"f":[11,24,-2,1],"c":1,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[11,24,-2,1],1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r01-polynomial-0100","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","language":"zh-Hant","source_mko_ids":["mko-polynomial"],"title_zh":"變數平移後的完整係數","stem_zh":"升冪係數 f=[12, 25, -2, 1]，c=2。求 f(x+c) 的完整升冪係數，不只求某點值。","answer":{"kind":"expression","value":"[62, 29, 4, 1]","explanation_zh":"平移後升冪係數=[62, 29, 4, 1]。用P←(x+c)P+a的Horner係數遞推，合併各次冪；常數不隨平移改變。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["變數平移後的完整係數"]},"variant":{"family_id":"qf-density10-poly-shift-coefficients","seed":202610101100,"parameters":{"f":[12,25,-2,1],"c":2,"representation":"ascending_powers","operation":"density10_poly_shift_coefficients","domain":"integer_polynomial_slice","semantic_template":"qf-density10-poly-shift-coefficients/[[12,25,-2,1],2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}