{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0001","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d15710fa2907e317c6cab40ae76c3994a572e6f227a5d6cb961c9c0bc0be91e","family_id":"qf-density10-poly-add-coefficients","seed":202610101001,"semantic_template":"qf-density10-poly-add-coefficients/[[0],[0]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[0] and g=[0], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[0]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [0]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0002","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cf1bf20bb67af38185f467cbd6f47e25f950f7bdbe0ec69227ef21ad1a18371","family_id":"qf-density10-poly-add-coefficients","seed":202610101002,"semantic_template":"qf-density10-poly-add-coefficients/[[-3],[2]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[2] and g=[-3], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0003","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37f2154fe64d04f7a540e585a5f0b434ee95d3a52af3ca98ed6f1a749d8fc8e0","family_id":"qf-density10-poly-add-coefficients","seed":202610101003,"semantic_template":"qf-density10-poly-add-coefficients/[[-3,2,-1],[3,-2,1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[3, -2, 1] and g=[-3, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[0]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [0]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0004","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab2f016fd6fd582525bebc31b4aa6c338204770eea0badf8e5d6beb1faedef43","family_id":"qf-density10-poly-add-coefficients","seed":202610101004,"semantic_template":"qf-density10-poly-add-coefficients/[[-9,4,-1,2],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-9, 4, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-8, 2, 1, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-8, 2, 1, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0005","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebb816cd16b413be06eed582b82ba0537c6fddfdac1e6014d9d35e38a5e20740","family_id":"qf-density10-poly-add-coefficients","seed":202610101005,"semantic_template":"qf-density10-poly-add-coefficients/[[-8,5,0,1],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-8, 5, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-7, 3, 2]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-7, 3, 2]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0006","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e4c2125026bcfb99b49d4a5b1107a296fb830735426eec438580fb8df5bbe2d","family_id":"qf-density10-poly-add-coefficients","seed":202610101006,"semantic_template":"qf-density10-poly-add-coefficients/[[-7,6,1,2],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-7, 6, 1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-6, 4, 3, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-6, 4, 3, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0007","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d8f206383c13d69fb253a4fe87e5b898c3b67b470b6beb985c480df8f889635","family_id":"qf-density10-poly-add-coefficients","seed":202610101007,"semantic_template":"qf-density10-poly-add-coefficients/[[-6,7,-1,1],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-6, 7, -1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-5, 5, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-5, 5, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0008","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a93c3c2f4c408ddfc3459382c201a1092036d85339da2af894897a29b4f47848","family_id":"qf-density10-poly-add-coefficients","seed":202610101008,"semantic_template":"qf-density10-poly-add-coefficients/[[-5,8,0,2],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-5, 8, 0, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-4, 6, 2, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-4, 6, 2, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0009","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58392080972d9ca7e8db22c4642f7b5efb91a49f12ca10ffcab74c87b690f518","family_id":"qf-density10-poly-add-coefficients","seed":202610101009,"semantic_template":"qf-density10-poly-add-coefficients/[[-4,9,1,1],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-4, 9, 1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-3, 7, 3]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-3, 7, 3]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0010","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5449021e4b924ecb062e6fbea9970ec497a7ae01ace347517a09d5f54b747953","family_id":"qf-density10-poly-add-coefficients","seed":202610101010,"semantic_template":"qf-density10-poly-add-coefficients/[[-3,10,-1,2],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-3, 10, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-2, 8, 1, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-2, 8, 1, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0011","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eed5b5d37b6e478c59c5a732de71ec93b233e59c2e53b5378f3f33fbbec41952","family_id":"qf-density10-poly-add-coefficients","seed":202610101011,"semantic_template":"qf-density10-poly-add-coefficients/[[-2,11,0,1],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-2, 11, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[-1, 9, 2]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-1, 9, 2]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0012","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e0d4e8433ef66268c8e653697574fe7bb4649c7b31949df6891262675868ae9","family_id":"qf-density10-poly-add-coefficients","seed":202610101012,"semantic_template":"qf-density10-poly-add-coefficients/[[-1,12,1,2],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[-1, 12, 1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[0, 10, 3, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [0, 10, 3, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0013","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9844457fe876abca1e7b92b164e4e33566ec38708bc39c17dda56086b22c983","family_id":"qf-density10-poly-add-coefficients","seed":202610101013,"semantic_template":"qf-density10-poly-add-coefficients/[[0,13,-1,1],[1,-2,2,-1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[0, 13, -1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[1, 11, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [1, 11, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0014","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ed6978a61321da17b257d2c0211c80c6cdc218aa4d0134341ed5ae4f2a06107","family_id":"qf-density10-poly-add-coefficients","seed":202610101014,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[1,14,0,2]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[1, 14, 0, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[2, 12, 2, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [2, 12, 2, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0015","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6fa07c1ddf09912612530104ca4f736adf8128acd5881a29bdaad44f891c2f7","family_id":"qf-density10-poly-add-coefficients","seed":202610101015,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[2,15,1,1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[2, 15, 1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[3, 13, 3]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [3, 13, 3]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0016","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88c66a9759eccaf1373cc5552977a9d30a1cb77fbdee4f4a20e718b4c76edc6b","family_id":"qf-density10-poly-add-coefficients","seed":202610101016,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[3,16,-1,2]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[3, 16, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[4, 14, 1, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [4, 14, 1, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0017","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6fa79bed493bc787a627e65d413dd461e0ff26b511dd4d68ec7a810c01d4cff","family_id":"qf-density10-poly-add-coefficients","seed":202610101017,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[4,17,0,1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[4, 17, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[5, 15, 2]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [5, 15, 2]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0018","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e1ea4541a6fc2489e5465adb0a67ea70e62d58e658c0eb001e5b72d22120fb9","family_id":"qf-density10-poly-add-coefficients","seed":202610101018,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[5,18,1,2]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[5, 18, 1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[6, 16, 3, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [6, 16, 3, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0019","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccd2b42ff84e181e63ab453c121d1049dfc10a2245160042489d74b805fb495b","family_id":"qf-density10-poly-add-coefficients","seed":202610101019,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[6,19,-1,1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[6, 19, -1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[7, 17, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [7, 17, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0020","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"322769a67ff2e99f89f5c9ed5870025b64689e9a6805ecd41f9c67dd3993a291","family_id":"qf-density10-poly-add-coefficients","seed":202610101020,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[7,20,0,2]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[7, 20, 0, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[8, 18, 2, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [8, 18, 2, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0021","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06b1e0b21570561f22f94fa31ac8675af58b70ce7669f43bdfbbcdb68baeb726","family_id":"qf-density10-poly-add-coefficients","seed":202610101021,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[8,21,1,1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[8, 21, 1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[9, 19, 3]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [9, 19, 3]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0022","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa8858a590e3fdb62ffa9f40a40dcd6389a7e8cb97732b04a93ea99c2977abc8","family_id":"qf-density10-poly-add-coefficients","seed":202610101022,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[9,22,-1,2]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[9, 22, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[10, 20, 1, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [10, 20, 1, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0023","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2abc43e73afdc74e0b61e881ec1f192f11a449146066253332d2461d131cdbd7","family_id":"qf-density10-poly-add-coefficients","seed":202610101023,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[10,23,0,1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[10, 23, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[11, 21, 2]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [11, 21, 2]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0024","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7fa779e82e6214b5353ca68ba9f9a6c572774b2ae42b6d29d30a11788a94e54","family_id":"qf-density10-poly-add-coefficients","seed":202610101024,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[11,24,1,2]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[11, 24, 1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[12, 22, 3, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [12, 22, 3, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0025","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b08de5f2498c40c7cfb160eea747a8c495926c38e6e75bb380937fb6743f0d0d","family_id":"qf-density10-poly-add-coefficients","seed":202610101025,"semantic_template":"qf-density10-poly-add-coefficients/[[1,-2,2,-1],[12,25,-1,1]]"},"fields":{"title":"Complete Polynomial Sum Coefficients","stem":"Integer polynomials have ascending coefficients f=[12, 25, -1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.","answer":{"kind":"expression","value":"[13, 23, 1]","explanation":"Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [13, 23, 1]."},"skills":["Complete Polynomial Sum Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0026","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"010fc79f540b98e680b2b1c8fd576122b2bba2f18aced4bddd653cdf12027885","family_id":"qf-density10-poly-product-coefficients","seed":202610101026,"semantic_template":"qf-density10-poly-product-coefficients/[\"zero_product\"]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[0] and g=[1, -1], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[0]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [0]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0027","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f04997fc7141b88bd485b43ac01ee8c0873f1868bf84a0ccf53ab3bc1d6cb77a","family_id":"qf-density10-poly-product-coefficients","seed":202610101027,"semantic_template":"qf-density10-poly-product-coefficients/[[-3],[2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[2] and g=[-3], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-6]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-6]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0028","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cc775effdc5e02ece9b8f7cf536ef14ab546b86b913482c40afdb9b5caf200e","family_id":"qf-density10-poly-product-coefficients","seed":202610101028,"semantic_template":"qf-density10-poly-product-coefficients/[[1,-1],[1,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[1, 1] and g=[1, -1], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[1, 0, -1]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [1, 0, -1]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0029","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0d32d06890b043baa960435b218d6e0ae200f559171f4ad2cc40d3e611a4b59","family_id":"qf-density10-poly-product-coefficients","seed":202610101029,"semantic_template":"qf-density10-poly-product-coefficients/[[-9,4,-3,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-9, 4, -3, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-18, 17, -19, -9, 4, -5, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-18, 17, -19, -9, 4, -5, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0030","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75e678c6ab4cd973db313c08b5a0e4d6e904860702681df687e70fb6c5c85f7f","family_id":"qf-density10-poly-product-coefficients","seed":202610101030,"semantic_template":"qf-density10-poly-product-coefficients/[[-8,5,-4,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-8, 5, -4, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-16, 18, -21, -5, 5, -7, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-16, 18, -21, -5, 5, -7, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0031","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be1a23338dbb2b42271ad4380cddcbbe62bccfa044c07fa4dac736f55dfca60b","family_id":"qf-density10-poly-product-coefficients","seed":202610101031,"semantic_template":"qf-density10-poly-product-coefficients/[[-7,6,-5,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-7, 6, -5, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-14, 19, -23, -1, 6, -9, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-14, 19, -23, -1, 6, -9, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0032","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d1cbacbe1b9a931eb901c50cb400c47b500da2fd602c7f344b7c49cde9db278","family_id":"qf-density10-poly-product-coefficients","seed":202610101032,"semantic_template":"qf-density10-poly-product-coefficients/[[-6,7,-6,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-6, 7, -6, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-12, 20, -25, 3, 7, -11, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-12, 20, -25, 3, 7, -11, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0033","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0b857a9ff2d2b6c4dae10c6695cf0f349ead767189675b4cd63ff863e9e0898","family_id":"qf-density10-poly-product-coefficients","seed":202610101033,"semantic_template":"qf-density10-poly-product-coefficients/[[-5,8,-7,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-5, 8, -7, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-10, 21, -27, 7, 8, -13, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-10, 21, -27, 7, 8, -13, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0034","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f091e8308f32fe307a164a4c31716144940cafc34eb658f4ebbf93dc3c56ecba","family_id":"qf-density10-poly-product-coefficients","seed":202610101034,"semantic_template":"qf-density10-poly-product-coefficients/[[-4,9,-8,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-4, 9, -8, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-8, 22, -29, 11, 9, -15, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-8, 22, -29, 11, 9, -15, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0035","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"211f67ba490c4d4f2e9ab2330d4e7783e71cd253b141f52b4eeb290db3b135ba","family_id":"qf-density10-poly-product-coefficients","seed":202610101035,"semantic_template":"qf-density10-poly-product-coefficients/[[-3,10,-9,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-3, 10, -9, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-6, 23, -31, 15, 10, -17, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-6, 23, -31, 15, 10, -17, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0036","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"503b902ca7c0f365a359a9197a4f7193091eafdcc156283ed7601350dbe56c16","family_id":"qf-density10-poly-product-coefficients","seed":202610101036,"semantic_template":"qf-density10-poly-product-coefficients/[[-2,11,-10,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-2, 11, -10, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-4, 24, -33, 19, 11, -19, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-4, 24, -33, 19, 11, -19, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0037","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1d2ebd6c1e99ac68f3385a4d8322981b3efd76195ab7ffc20f503bf34e4830a","family_id":"qf-density10-poly-product-coefficients","seed":202610101037,"semantic_template":"qf-density10-poly-product-coefficients/[[-1,12,-11,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[-1, 12, -11, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[-2, 25, -35, 23, 12, -21, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-2, 25, -35, 23, 12, -21, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0038","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c134a59e3ed0e008f559ebbcdfc65841bdb1dc0d2632823d308ce2a8d38864a","family_id":"qf-density10-poly-product-coefficients","seed":202610101038,"semantic_template":"qf-density10-poly-product-coefficients/[[0,13,-12,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[0, 13, -12, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[0, 26, -37, 27, 13, -23, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [0, 26, -37, 27, 13, -23, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0039","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3546a9bb6e4c5b9476f24c6fe94f9deadc33f16b6a96b44cb6e4cf66695d365","family_id":"qf-density10-poly-product-coefficients","seed":202610101039,"semantic_template":"qf-density10-poly-product-coefficients/[[1,14,-13,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[1, 14, -13, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[2, 27, -39, 31, 14, -25, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [2, 27, -39, 31, 14, -25, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0040","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"280026116bb020993a8b9d6c2e65ac3a9970da07f46676f3cb7a319dc0677ee1","family_id":"qf-density10-poly-product-coefficients","seed":202610101040,"semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[2,15,-14,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[2, 15, -14, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[4, 28, -41, 35, 15, -27, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [4, 28, -41, 35, 15, -27, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0041","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b7e42b29cf04a5bbee5943b118d0340798ed640101a5a3246b866abed1f10f5","family_id":"qf-density10-poly-product-coefficients","seed":202610101041,"semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[3,16,-15,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[3, 16, -15, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[6, 29, -43, 39, 16, -29, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [6, 29, -43, 39, 16, -29, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0042","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2271af7a866cc1097f49466005c9af3ac9007a6ea4aa5ccf4985550e4b8b27ad","family_id":"qf-density10-poly-product-coefficients","seed":202610101042,"semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[4,17,-16,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[4, 17, -16, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[8, 30, -45, 43, 17, -31, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [8, 30, -45, 43, 17, -31, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0043","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d722ff0809cd8d8cc24d97564e6195a7b1754667f39357a3a0d846acb9b3c9d1","family_id":"qf-density10-poly-product-coefficients","seed":202610101043,"semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[5,18,-17,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[5, 18, -17, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[10, 31, -47, 47, 18, -33, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [10, 31, -47, 47, 18, -33, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0044","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1428b5b56a91cedd77b90757621d2434f4596ef4343696297dc87666c4d71897","family_id":"qf-density10-poly-product-coefficients","seed":202610101044,"semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[6,19,-18,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[6, 19, -18, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[12, 32, -49, 51, 19, -35, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [12, 32, -49, 51, 19, -35, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0045","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af9b101ea773ab33076c551a514a11b67c693bc6f4becee6dbaeb93d7530eef2","family_id":"qf-density10-poly-product-coefficients","seed":202610101045,"semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[7,20,-19,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[7, 20, -19, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[14, 33, -51, 55, 20, -37, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [14, 33, -51, 55, 20, -37, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0046","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3c6039e056d4b30e68b0451e73c8c09eab7d401c427718726b4148e42f4e850","family_id":"qf-density10-poly-product-coefficients","seed":202610101046,"semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[8,21,-20,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[8, 21, -20, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[16, 34, -53, 59, 21, -39, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [16, 34, -53, 59, 21, -39, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0047","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d0434ba0bd9a5a55a25c2c998bdabbf043ae159f2630d64d3f170dfd67c5124","family_id":"qf-density10-poly-product-coefficients","seed":202610101047,"semantic_template":"qf-density10-poly-product-coefficients/[[2,-1,1,2],[9,22,-21,1]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[9, 22, -21, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[18, 35, -55, 63, 22, -41, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [18, 35, -55, 63, 22, -41, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0048","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9574bd50e83fd39bfdbadd7f41689eb24f1971222225ca0fffc94ae5d431b1e","family_id":"qf-density10-poly-product-coefficients","seed":202610101048,"semantic_template":"qf-density10-poly-product-coefficients/[[10,23,-22,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[10, 23, -22, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[20, 36, -57, 67, 23, -43, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [20, 36, -57, 67, 23, -43, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0049","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f18d3948a6d74fc4fa91bdb215edb9702743d1094b94b55d453dd3a106da78a1","family_id":"qf-density10-poly-product-coefficients","seed":202610101049,"semantic_template":"qf-density10-poly-product-coefficients/[[11,24,-23,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[11, 24, -23, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[22, 37, -59, 71, 24, -45, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [22, 37, -59, 71, 24, -45, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0050","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccb71f014b681ad851896d8197d26d444e6d7b96106a4ebbf84acd4f1c297319","family_id":"qf-density10-poly-product-coefficients","seed":202610101050,"semantic_template":"qf-density10-poly-product-coefficients/[[12,25,-24,1],[2,-1,1,2]]"},"fields":{"title":"Complete Polynomial Product Coefficients","stem":"Integer polynomials have ascending coefficients f=[12, 25, -24, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.","answer":{"kind":"expression","value":"[24, 38, -61, 75, 25, -47, 2]","explanation":"Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [24, 38, -61, 75, 25, -47, 2]."},"skills":["Complete Polynomial Product Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0051","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0d0948f4dcdf0c1e27b8280ff7a37129de484f2fec62497e7fc61c1761dc330","family_id":"qf-density10-poly-two-point-values","seed":202610101051,"semantic_template":"qf-density10-poly-two-point-values/[[0]]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[0], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (0, 0)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0052","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5de9e05608fc980c5bb260c0394ef4811fe40a090dee22229c32b1cea7da6ac0","family_id":"qf-density10-poly-two-point-values","seed":202610101052,"semantic_template":"qf-density10-poly-two-point-values/[[-3]]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-3], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-3, -3)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-3, -3)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0053","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d07f27f45deb0bcc67bb5e97f37ec3455824e3abbee88b42057946ff253f80f","family_id":"qf-density10-poly-two-point-values","seed":202610101053,"semantic_template":"qf-density10-poly-two-point-values/[[-10,2,-1,2],-2,3]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-10, 2, -1, 2], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-34, 41)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-34, 41)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0054","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14aa4c044f974b024cef1d1ce1b464db06113f885515ce6405ac3a895f581f82","family_id":"qf-density10-poly-two-point-values","seed":202610101054,"semantic_template":"qf-density10-poly-two-point-values/[[-9,2,-1],-2,4]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-9, 2, -1, 0], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-17, -17)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-17, -17)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0055","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7af1db94e0538e82d96aa2179b84fa7bf0f5127f74e62105c533c3c16ba20e6a","family_id":"qf-density10-poly-two-point-values","seed":202610101055,"semantic_template":"qf-density10-poly-two-point-values/[[-8,2,-1,1],-2,1]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-8, 2, -1, 1], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-24, -6)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-24, -6)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0056","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"473f91b512a961e9ab7dd5502874a4caea88bc4fb762443f6cbe7a57940912a8","family_id":"qf-density10-poly-two-point-values","seed":202610101056,"semantic_template":"qf-density10-poly-two-point-values/[[-7,2,-1,2],-2,2]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-7, 2, -1, 2], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-31, 9)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-31, 9)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0057","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c67942f973206924386f9a7cf079ed26416163d4524feac16ce585b9be47b92","family_id":"qf-density10-poly-two-point-values","seed":202610101057,"semantic_template":"qf-density10-poly-two-point-values/[[-6,2,-1],-2,3]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-6, 2, -1, 0], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-14, -9)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-14, -9)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0058","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4fde32f7ebbee81429812f0f92240e9b272963f5728d8a7442a03f54426ee0a","family_id":"qf-density10-poly-two-point-values","seed":202610101058,"semantic_template":"qf-density10-poly-two-point-values/[[-5,2,-1,1],-2,4]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-5, 2, -1, 1], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-21, 51)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-21, 51)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0059","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8247e266f285707009fb4adecf022ea1374f1754c4eed9c6adad0e0c455382cf","family_id":"qf-density10-poly-two-point-values","seed":202610101059,"semantic_template":"qf-density10-poly-two-point-values/[[-4,2,-1,2],-2,1]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-4, 2, -1, 2], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-28, -1)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-28, -1)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0060","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10fd639624ab6ab85efbd301cbe7f68075386c450fdca34725361d7d3fe3eeff","family_id":"qf-density10-poly-two-point-values","seed":202610101060,"semantic_template":"qf-density10-poly-two-point-values/[[-3,2,-1],-2,2]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-3, 2, -1, 0], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-11, -3)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-11, -3)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0061","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18df418230c55b7f0900c9f7ff2652cbcf04beeab666a6aec9be829f2c39b2a4","family_id":"qf-density10-poly-two-point-values","seed":202610101061,"semantic_template":"qf-density10-poly-two-point-values/[[-2,2,-1,1],-2,3]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-2, 2, -1, 1], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-18, 22)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-18, 22)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0062","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0fc73bde8b5895c546576f4e4b8c8f1be7db80bbc996f6dbddebd6c095355cd","family_id":"qf-density10-poly-two-point-values","seed":202610101062,"semantic_template":"qf-density10-poly-two-point-values/[[-1,2,-1,2],-2,4]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[-1, 2, -1, 2], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-25, 119)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-25, 119)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0063","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86acd1173a1df228252a7fd95234e4dcd68fa323ec8020849fa419b300969310","family_id":"qf-density10-poly-two-point-values","seed":202610101063,"semantic_template":"qf-density10-poly-two-point-values/[[0,2,-1],-2,1]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[0, 2, -1, 0], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-8, 1)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-8, 1)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0064","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20b110eb39ddc75117786fec95f03ee260a9cd6f33fdb0e92afb54dfae2e54b6","family_id":"qf-density10-poly-two-point-values","seed":202610101064,"semantic_template":"qf-density10-poly-two-point-values/[[1,2,-1,1],-2,2]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[1, 2, -1, 1], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-15, 9)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-15, 9)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0065","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f32911450a66f684155cdf09cfd3c52221217c4e1f2a1f1a1a255e9bf7c06db","family_id":"qf-density10-poly-two-point-values","seed":202610101065,"semantic_template":"qf-density10-poly-two-point-values/[[2,2,-1,2],-2,3]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[2, 2, -1, 2], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-22, 53)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-22, 53)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0066","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b093bfbcc541ca67f779d5e6e08e17340e5440f188e4d872937918fe9ccbe61","family_id":"qf-density10-poly-two-point-values","seed":202610101066,"semantic_template":"qf-density10-poly-two-point-values/[[3,2,-1],-2,4]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[3, 2, -1, 0], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-5, -5)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-5, -5)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0067","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97fef78cb0fe06ebfbdca71a6124e38669f2618568178ac8f187fb4bdf4d7f7c","family_id":"qf-density10-poly-two-point-values","seed":202610101067,"semantic_template":"qf-density10-poly-two-point-values/[[4,2,-1,1],-2,1]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[4, 2, -1, 1], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-12, 6)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-12, 6)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0068","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bbd195b239cf27d54f00b6c9de6f3571956e0340ff7e4f688f0bfcac6c11113","family_id":"qf-density10-poly-two-point-values","seed":202610101068,"semantic_template":"qf-density10-poly-two-point-values/[[5,2,-1,2],-2,2]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[5, 2, -1, 2], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-19, 21)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-19, 21)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0069","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35a8cb12ab6c811890ad4e05a35b611d3388eb158537dd65e1c59c42d38891dc","family_id":"qf-density10-poly-two-point-values","seed":202610101069,"semantic_template":"qf-density10-poly-two-point-values/[[6,2,-1],-2,3]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[6, 2, -1, 0], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-2, 3)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-2, 3)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0070","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac79b9f4a653a8d439b293fc69fb4b560a0aa604e0c866ca04e1086a16d8991d","family_id":"qf-density10-poly-two-point-values","seed":202610101070,"semantic_template":"qf-density10-poly-two-point-values/[[7,2,-1,1],-2,4]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[7, 2, -1, 1], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-9, 63)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-9, 63)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0071","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bcb8dfa4d9b80f295153b5baccadd0948917dcc5fa6e47865839dbf373f308e","family_id":"qf-density10-poly-two-point-values","seed":202610101071,"semantic_template":"qf-density10-poly-two-point-values/[[8,2,-1,2],-2,1]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[8, 2, -1, 2], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-16, 11)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-16, 11)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0072","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32a1d2b017abb11383700cb33be763a104bbf04de2dad0811f0a1938dad0e018","family_id":"qf-density10-poly-two-point-values","seed":202610101072,"semantic_template":"qf-density10-poly-two-point-values/[[9,2,-1],-2,2]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[9, 2, -1, 0], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(1, 9)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (1, 9)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0073","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8438fb8467e8b5be4e344aa7108945ea3104105ca05023d9ce76bc264517bf7","family_id":"qf-density10-poly-two-point-values","seed":202610101073,"semantic_template":"qf-density10-poly-two-point-values/[[10,2,-1,1],-2,3]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[10, 2, -1, 1], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-6, 34)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-6, 34)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0074","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"193664d9cb257cf3a8e70b8e77a64139d75be87cc4c6a25bceff92dcc1023b9b","family_id":"qf-density10-poly-two-point-values","seed":202610101074,"semantic_template":"qf-density10-poly-two-point-values/[[11,2,-1,2],-2,4]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[11, 2, -1, 2], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(-13, 131)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-13, 131)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0075","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9194357dfd2d568f6755b19420208733b7844d0ffe5af60cd56aad15013317f3","family_id":"qf-density10-poly-two-point-values","seed":202610101075,"semantic_template":"qf-density10-poly-two-point-values/[[12,2,-1],-2,1]"},"fields":{"title":"Ordered Polynomial Values at Two Points","stem":"Integer polynomials have ascending coefficients f=[12, 2, -1, 0], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.","answer":{"kind":"expression","value":"(4, 13)","explanation":"Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (4, 13)."},"skills":["Ordered Polynomial Values at Two Points"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0076","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97244fc2ff3395ca8966381a4d3cdcd2b588124ed7b3a08e6c2e24933d947b8e","family_id":"qf-density10-poly-shift-coefficients","seed":202610101076,"semantic_template":"qf-density10-poly-shift-coefficients/[[0]]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[0], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[0]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [0]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0077","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d3911fad617bf167ac25463c49acb5f0063a8690f17e1ee0eaa4ea6a6776ffb","family_id":"qf-density10-poly-shift-coefficients","seed":202610101077,"semantic_template":"qf-density10-poly-shift-coefficients/[[-4]]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-4], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-4]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-4]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0078","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ba0ee793096b1934dbeb5b3f426f1997a31d7846e7e21b6610033240571c741","family_id":"qf-density10-poly-shift-coefficients","seed":202610101078,"semantic_template":"qf-density10-poly-shift-coefficients/[[-10,3,-2,1],0]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-10, 3, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-10, 3, -2, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-10, 3, -2, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0079","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b749b7164650fc379b6311d8b0528b936dc0b87589e3138cdaa919bdef86b997","family_id":"qf-density10-poly-shift-coefficients","seed":202610101079,"semantic_template":"qf-density10-poly-shift-coefficients/[[-9,4,-2,1],1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-9, 4, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-6, 3, 1, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-6, 3, 1, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0080","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87587cb134f31d6ecc017dd3a7834fc3d03fdc21fa644e78a50e126482323aee","family_id":"qf-density10-poly-shift-coefficients","seed":202610101080,"semantic_template":"qf-density10-poly-shift-coefficients/[[-8,5,-2,1],2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-8, 5, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[2, 9, 4, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [2, 9, 4, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0081","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9dbf74edf66095ec4db8065d4f8919d862609955ebad237298d3fac2a8a8af7","family_id":"qf-density10-poly-shift-coefficients","seed":202610101081,"semantic_template":"qf-density10-poly-shift-coefficients/[[-7,6,-2,1],-2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-7, 6, -2, 1], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-35, 26, -8, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-35, 26, -8, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0082","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a7bb0c61ec7c74124541c827ce6c8bfa67e03ad448a5506dfd74bdb6af9919a","family_id":"qf-density10-poly-shift-coefficients","seed":202610101082,"semantic_template":"qf-density10-poly-shift-coefficients/[[-6,7,-2,1],-1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-6, 7, -2, 1], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-16, 14, -5, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-16, 14, -5, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0083","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e21d022f7e2b074c828f71b95132bdb4559a680f64b037daf992a91ac873f82f","family_id":"qf-density10-poly-shift-coefficients","seed":202610101083,"semantic_template":"qf-density10-poly-shift-coefficients/[[-5,8,-2,1],0]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-5, 8, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-5, 8, -2, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-5, 8, -2, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0084","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8757b63b752d252ef97374d5a430bebb553acd864a350f843d8a6e8b1d12f0c","family_id":"qf-density10-poly-shift-coefficients","seed":202610101084,"semantic_template":"qf-density10-poly-shift-coefficients/[[-4,9,-2,1],1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-4, 9, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[4, 8, 1, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [4, 8, 1, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0085","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"569fd1c118eb032bb4632142559883b755d94c9edfb853720eae940d91ee8ada","family_id":"qf-density10-poly-shift-coefficients","seed":202610101085,"semantic_template":"qf-density10-poly-shift-coefficients/[[-3,10,-2,1],2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-3, 10, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[17, 14, 4, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [17, 14, 4, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0086","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c71d0fc237c9355f5e811cdbea49f1f8d71308985b50ac6112ef27df65ecbaf2","family_id":"qf-density10-poly-shift-coefficients","seed":202610101086,"semantic_template":"qf-density10-poly-shift-coefficients/[[-2,11,-2,1],-2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-2, 11, -2, 1], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-40, 31, -8, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-40, 31, -8, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0087","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"803ce23acabef5e3efca21dfb1e812b2e03cb02ae11d3e9537fd674cc8c46af8","family_id":"qf-density10-poly-shift-coefficients","seed":202610101087,"semantic_template":"qf-density10-poly-shift-coefficients/[[-1,12,-2,1],-1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[-1, 12, -2, 1], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-16, 19, -5, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-16, 19, -5, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0088","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39ba64834a7240cc2a5306642bd1351d75e5dd54805c23cc113c80ebd7d07a30","family_id":"qf-density10-poly-shift-coefficients","seed":202610101088,"semantic_template":"qf-density10-poly-shift-coefficients/[[0,13,-2,1],0]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[0, 13, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[0, 13, -2, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [0, 13, -2, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0089","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51c195bb52fd5c9be09f504d8152be6250fba700440deff2d6f967780dab8143","family_id":"qf-density10-poly-shift-coefficients","seed":202610101089,"semantic_template":"qf-density10-poly-shift-coefficients/[[1,14,-2,1],1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[1, 14, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[14, 13, 1, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [14, 13, 1, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0090","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edf102306ba7378507ee58d0caa9b4de19beaaeadfadc5cf494a4e5e617a2a3c","family_id":"qf-density10-poly-shift-coefficients","seed":202610101090,"semantic_template":"qf-density10-poly-shift-coefficients/[[2,15,-2,1],2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[2, 15, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[32, 19, 4, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [32, 19, 4, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0091","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b870f0cc48bd32fa66d927558bd0cfce326081d954a0fe89eb0029f74a6b934","family_id":"qf-density10-poly-shift-coefficients","seed":202610101091,"semantic_template":"qf-density10-poly-shift-coefficients/[[3,16,-2,1],-2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[3, 16, -2, 1], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-45, 36, -8, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-45, 36, -8, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0092","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f8132070a06dc5fcc89d9e3f9d57d5cd8e9b69d6c7705d531d7ebbf88fd4eef","family_id":"qf-density10-poly-shift-coefficients","seed":202610101092,"semantic_template":"qf-density10-poly-shift-coefficients/[[4,17,-2,1],-1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[4, 17, -2, 1], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-16, 24, -5, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-16, 24, -5, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0093","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdf2ba0ea5ba374f867196a5ab63e9c33e1527b2bcd856bbbbdf30cfce2cb229","family_id":"qf-density10-poly-shift-coefficients","seed":202610101093,"semantic_template":"qf-density10-poly-shift-coefficients/[[5,18,-2,1],0]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[5, 18, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[5, 18, -2, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [5, 18, -2, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0094","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f4acd44d6d46056186c2954c7ebd9ea0b8e4206b7858811919253a8afe76ac7","family_id":"qf-density10-poly-shift-coefficients","seed":202610101094,"semantic_template":"qf-density10-poly-shift-coefficients/[[6,19,-2,1],1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[6, 19, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[24, 18, 1, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [24, 18, 1, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0095","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"343a33a72afe9fc67626327f6babc7f713b0069c3df242b5def0ed0772b03e1a","family_id":"qf-density10-poly-shift-coefficients","seed":202610101095,"semantic_template":"qf-density10-poly-shift-coefficients/[[7,20,-2,1],2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[7, 20, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[47, 24, 4, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [47, 24, 4, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0096","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3dc88f94e376913a6e623c139292cb684a5aafb421b046586c522fa1b0abe44","family_id":"qf-density10-poly-shift-coefficients","seed":202610101096,"semantic_template":"qf-density10-poly-shift-coefficients/[[8,21,-2,1],-2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[8, 21, -2, 1], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-50, 41, -8, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-50, 41, -8, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0097","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8488426dbac3e26b1e347c055d0748ef667d6bf8187447e0cbc27b4be8a98cc","family_id":"qf-density10-poly-shift-coefficients","seed":202610101097,"semantic_template":"qf-density10-poly-shift-coefficients/[[9,22,-2,1],-1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[9, 22, -2, 1], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[-16, 29, -5, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-16, 29, -5, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0098","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c39997ba793e8c9be2c4a0586d1f544e409c87e5ea2f247042b69be979687a8","family_id":"qf-density10-poly-shift-coefficients","seed":202610101098,"semantic_template":"qf-density10-poly-shift-coefficients/[[10,23,-2,1],0]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[10, 23, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[10, 23, -2, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [10, 23, -2, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0099","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ebdb190a875a91d95189fb53f92275d2e10221abf7b697cceb4ec52839edd9a","family_id":"qf-density10-poly-shift-coefficients","seed":202610101099,"semantic_template":"qf-density10-poly-shift-coefficients/[[11,24,-2,1],1]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[11, 24, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[34, 23, 1, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [34, 23, 1, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r01-polynomial-0100","batch_id":"batch-20261008-advance-20261010-r01-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7497cdff1876ae4f4344ec1e694d771b14b21fabdf0b1850ed1820d4bbffebd","family_id":"qf-density10-poly-shift-coefficients","seed":202610101100,"semantic_template":"qf-density10-poly-shift-coefficients/[[12,25,-2,1],2]"},"fields":{"title":"Complete Polynomial Translation Coefficients","stem":"Integer polynomials have ascending coefficients f=[12, 25, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.","answer":{"kind":"expression","value":"[62, 29, 4, 1]","explanation":"Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [62, 29, 4, 1]."},"skills":["Complete Polynomial Translation Coefficients"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}