{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r03-factorization-0001","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b4aee1f83af03eec273b0755c872a37ffe6ed1d493c0e75480a1855b9ce9cf4","family_id":"qf-factorization-product","seed":202610033001,"semantic_template":"qf-factorization-product/{\"left\":[\"7\",\"1\"],\"operation\":\"product\",\"right\":[]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [0] and [7, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[0]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [0]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0002","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b58f2512233acb176334c953afe6021c1720a6cbb19316fa7693c6eeda380b12","family_id":"qf-factorization-product","seed":202610033002,"semantic_template":"qf-factorization-product/{\"left\":[\"-11\",\"1\"],\"operation\":\"product\",\"right\":[\"3\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-11, 1] and [3, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-33, -8, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-33, -8, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0003","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d35db61d48ee1951dcf6ef077f16382f9957eb86ca203385bc2830d457156d3","family_id":"qf-factorization-product","seed":202610033003,"semantic_template":"qf-factorization-product/{\"left\":[\"-10\",\"1\"],\"operation\":\"product\",\"right\":[\"4\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-10, 1] and [4, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-40, -6, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-40, -6, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0004","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b9120209791fddd386ec56e1e406078845b8f5197a6dff1889c4aa2bcd340e5","family_id":"qf-factorization-product","seed":202610033004,"semantic_template":"qf-factorization-product/{\"left\":[\"-9\",\"1\"],\"operation\":\"product\",\"right\":[\"5\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-9, 1] and [5, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-45, -4, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-45, -4, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0005","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddb126b3d6108722b4d913b2142c286c14dd02d53a78792e25828122329d5830","family_id":"qf-factorization-product","seed":202610033005,"semantic_template":"qf-factorization-product/{\"left\":[\"-8\",\"1\"],\"operation\":\"product\",\"right\":[\"6\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-8, 1] and [6, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-48, -2, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-48, -2, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0006","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1c35d3089c06f328acfa3986691e966ac1d8375038532c73b62559e2bedb3d5","family_id":"qf-factorization-product","seed":202610033006,"semantic_template":"qf-factorization-product/{\"left\":[\"-7\",\"1\"],\"operation\":\"product\",\"right\":[\"7\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-7, 1] and [7, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-49, 0, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-49, 0, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0007","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d339e90c5a8b4cf00e635e8e9cd1dea5b7eb3476d537f7853129e7c8d41c1f4c","family_id":"qf-factorization-product","seed":202610033007,"semantic_template":"qf-factorization-product/{\"left\":[\"-6\",\"1\"],\"operation\":\"product\",\"right\":[\"8\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-6, 1] and [8, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-48, 2, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-48, 2, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0008","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db75a553c25fb68e6db8e56d2f6c059641c49ee6c0c9546210384f0b692036a3","family_id":"qf-factorization-product","seed":202610033008,"semantic_template":"qf-factorization-product/{\"left\":[\"-5\",\"1\"],\"operation\":\"product\",\"right\":[\"9\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-5, 1] and [9, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-45, 4, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-45, 4, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0009","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9692a97371378906bb34ae19ce28b8e80f869bc63dd7eceab635972c5f0d3e8f","family_id":"qf-factorization-product","seed":202610033009,"semantic_template":"qf-factorization-product/{\"left\":[\"-4\",\"1\"],\"operation\":\"product\",\"right\":[\"10\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-4, 1] and [10, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-40, 6, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-40, 6, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0010","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb031a5801ea348c7dd21dc8e57679571814fb406c802aefde68e9e3b82d539f","family_id":"qf-factorization-product","seed":202610033010,"semantic_template":"qf-factorization-product/{\"left\":[\"-3\",\"1\"],\"operation\":\"product\",\"right\":[\"11\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-3, 1] and [11, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-33, 8, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-33, 8, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0011","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd0dd7f1431b97cf4420e622c57dd01c3402509e4f0be48bf941b619050b6501","family_id":"qf-factorization-product","seed":202610033011,"semantic_template":"qf-factorization-product/{\"left\":[\"-2\",\"1\"],\"operation\":\"product\",\"right\":[\"12\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-2, 1] and [12, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-24, 10, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-24, 10, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0012","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bc952999e08e31f1eda744794bc92dd1705d314094ecc435c2b4b15ff2cb1df","family_id":"qf-factorization-product","seed":202610033012,"semantic_template":"qf-factorization-product/{\"left\":[\"-1\",\"1\"],\"operation\":\"product\",\"right\":[\"13\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-1, 1] and [13, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[-13, 12, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-13, 12, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0013","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9e077e29ee03b61f96af8f5c66d95a6bd929825c8f413777df8ffb4a079658e","family_id":"qf-factorization-product","seed":202610033013,"semantic_template":"qf-factorization-product/{\"left\":[\"0\",\"1\"],\"operation\":\"product\",\"right\":[\"14\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [0, 1] and [14, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[0, 14, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [0, 14, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0014","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d210dc2aed289fba7856fa397e8a3e76ef310f06e1af774621a7a6fee0984cd","family_id":"qf-factorization-product","seed":202610033014,"semantic_template":"qf-factorization-product/{\"left\":[\"1\",\"1\"],\"operation\":\"product\",\"right\":[\"15\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [1, 1] and [15, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[15, 16, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [15, 16, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0015","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2dcd02b2345f72da75f90b6917ff44c1490f9452dac00c66b90c3957fbe33c2","family_id":"qf-factorization-product","seed":202610033015,"semantic_template":"qf-factorization-product/{\"left\":[\"16\",\"1\"],\"operation\":\"product\",\"right\":[\"2\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [2, 1] and [16, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[32, 18, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [32, 18, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0016","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89cfc0837ae9a8a0cf1180743ba47ad675aa45abef3231256d90690279ac615d","family_id":"qf-factorization-product","seed":202610033016,"semantic_template":"qf-factorization-product/{\"left\":[\"17\",\"1\"],\"operation\":\"product\",\"right\":[\"3\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [3, 1] and [17, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[51, 20, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [51, 20, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0017","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ab2fcab9df52ca55b9cb50963d49913d78681fa385547cd5e954c35452d6af2","family_id":"qf-factorization-product","seed":202610033017,"semantic_template":"qf-factorization-product/{\"left\":[\"18\",\"1\"],\"operation\":\"product\",\"right\":[\"4\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [4, 1] and [18, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[72, 22, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [72, 22, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0018","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46ccad3c89736e8eaebe0907824b9e89fa391067f7f284685403b4ad6f91f196","family_id":"qf-factorization-product","seed":202610033018,"semantic_template":"qf-factorization-product/{\"left\":[\"19\",\"1\"],\"operation\":\"product\",\"right\":[\"5\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [5, 1] and [19, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[95, 24, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [95, 24, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0019","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"131690d00e55a21d8279c52b1a9ce58b3718006ebb034f321b697e00da6f88ff","family_id":"qf-factorization-product","seed":202610033019,"semantic_template":"qf-factorization-product/{\"left\":[\"20\",\"1\"],\"operation\":\"product\",\"right\":[\"6\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [6, 1] and [20, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[120, 26, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [120, 26, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0020","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"671c8e2700dcef40ab3d8e07ddf488de454852d241200da485805b66d09e3bb9","family_id":"qf-factorization-product","seed":202610033020,"semantic_template":"qf-factorization-product/{\"left\":[\"21\",\"1\"],\"operation\":\"product\",\"right\":[\"7\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [7, 1] and [21, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[147, 28, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [147, 28, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0021","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0404908906cd81ab6d8ab87c697ef29a7a2e35c4c173043c02d41c60cdb4babb","family_id":"qf-factorization-product","seed":202610033021,"semantic_template":"qf-factorization-product/{\"left\":[\"22\",\"1\"],\"operation\":\"product\",\"right\":[\"8\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [8, 1] and [22, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[176, 30, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [176, 30, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0022","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c3be33dc0e64ff346713aaecbcda80d1ae36af191b87d15694194179ce7fafa","family_id":"qf-factorization-product","seed":202610033022,"semantic_template":"qf-factorization-product/{\"left\":[\"23\",\"1\"],\"operation\":\"product\",\"right\":[\"9\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [9, 1] and [23, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[207, 32, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [207, 32, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0023","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8676f2445a9d1a5d8c9cdf81c6afe43fbfa4c1760fab8d08f68861e26c48ae1c","family_id":"qf-factorization-product","seed":202610033023,"semantic_template":"qf-factorization-product/{\"left\":[\"10\",\"1\"],\"operation\":\"product\",\"right\":[\"24\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [10, 1] and [24, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[240, 34, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [240, 34, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0024","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba140ef8b49a809c9592a06470c7d07203e53a8082819ce19d17bd9eecaabee3","family_id":"qf-factorization-product","seed":202610033024,"semantic_template":"qf-factorization-product/{\"left\":[\"11\",\"1\"],\"operation\":\"product\",\"right\":[\"25\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [11, 1] and [25, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[275, 36, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [275, 36, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0025","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7da5443ec38fcf9c7c1e3dd9bac59ef94013de605691245a8ac5878ccec90ac5","family_id":"qf-factorization-product","seed":202610033025,"semantic_template":"qf-factorization-product/{\"left\":[\"12\",\"1\"],\"operation\":\"product\",\"right\":[\"26\",\"1\"]}"},"fields":{"title":"Complete Coefficients of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [12, 1] and [26, 1] and return the normalized complete coefficient vector.","answer":{"kind":"expression","value":"[312, 38, 1]","explanation":"Multiply term by term and collect coefficients with the same exponent sum. The complete product is [312, 38, 1]; the identity is determined by all coefficients rather than sampled values."},"skills":["Complete Coefficients of a Factor Product"],"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-20261003-r03-factorization-0026","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"704b892bbb41ccfe5c44948731145c6e8a4a7c468900fbf4eabbc45f2fe94ef6","family_id":"qf-factorization-identity","seed":202610033026,"semantic_template":"qf-factorization-identity/{\"left\":[\"7\",\"1\"],\"operation\":\"identity\",\"proposed\":[],\"right\":[]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [0] and [7, 1] equal to the proposed polynomial [0]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0027","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c2b8a70a04537a5a019d2bf03a09d94fe4c548a9b10831871f383f6581de0dc","family_id":"qf-factorization-identity","seed":202610033027,"semantic_template":"qf-factorization-identity/{\"left\":[\"-11\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-32\",\"-8\",\"1\"],\"right\":[\"3\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-11, 1] and [3, 1] equal to the proposed polynomial [-32, -8, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0028","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a7abc2df3d8faf2f0fbbd1d9dc09b49e3bd08c07434353ebfcf8c250bf54228","family_id":"qf-factorization-identity","seed":202610033028,"semantic_template":"qf-factorization-identity/{\"left\":[\"-10\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-40\",\"-6\",\"2\"],\"right\":[\"4\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-10, 1] and [4, 1] equal to the proposed polynomial [-40, -6, 2]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0029","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c3fa7cb85aa4a45f09649b4b21c5c5426590fe1cae91f4dd8d5ef816d4e16c5","family_id":"qf-factorization-identity","seed":202610033029,"semantic_template":"qf-factorization-identity/{\"left\":[\"-9\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-45\",\"-4\",\"1\"],\"right\":[\"5\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-9, 1] and [5, 1] equal to the proposed polynomial [-45, -4, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0030","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f5316eccb36f7c456f1c124ac1aa0cd05be18f93f58d343562645d9af929285","family_id":"qf-factorization-identity","seed":202610033030,"semantic_template":"qf-factorization-identity/{\"left\":[\"-8\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-47\",\"-2\",\"1\"],\"right\":[\"6\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-8, 1] and [6, 1] equal to the proposed polynomial [-47, -2, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0031","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4508f29b6a7316b6aacd18d3ff7c43ba29da998143950cc76ada4446ad7afb9","family_id":"qf-factorization-identity","seed":202610033031,"semantic_template":"qf-factorization-identity/{\"left\":[\"-7\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-49\",\"0\",\"2\"],\"right\":[\"7\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-7, 1] and [7, 1] equal to the proposed polynomial [-49, 0, 2]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0032","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71f5c06660097b5d51e5095ea8f31bcfcb4f3cd977d7e424aca09e918004250d","family_id":"qf-factorization-identity","seed":202610033032,"semantic_template":"qf-factorization-identity/{\"left\":[\"-6\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-48\",\"2\",\"1\"],\"right\":[\"8\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-6, 1] and [8, 1] equal to the proposed polynomial [-48, 2, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0033","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"540eae56555b083a798eef8645529482e63b1b69d851a50b9ab0561501b69623","family_id":"qf-factorization-identity","seed":202610033033,"semantic_template":"qf-factorization-identity/{\"left\":[\"-5\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-44\",\"4\",\"1\"],\"right\":[\"9\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-5, 1] and [9, 1] equal to the proposed polynomial [-44, 4, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0034","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a96477d7302f30d5736d0bc4bc2e19bc8c2dc0694c3860f0f0e872c333557a6","family_id":"qf-factorization-identity","seed":202610033034,"semantic_template":"qf-factorization-identity/{\"left\":[\"-4\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-40\",\"6\",\"2\"],\"right\":[\"10\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-4, 1] and [10, 1] equal to the proposed polynomial [-40, 6, 2]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0035","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0d4416251e8de282aea82ae2e204163efa8ca3e7299e3995ae9feb10f1e56fb","family_id":"qf-factorization-identity","seed":202610033035,"semantic_template":"qf-factorization-identity/{\"left\":[\"-3\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-33\",\"8\",\"1\"],\"right\":[\"11\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-3, 1] and [11, 1] equal to the proposed polynomial [-33, 8, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0036","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ea7405178b5dcb6e5b6a0ccf765548ad81b43688723e5cb3dae8a693be056cc","family_id":"qf-factorization-identity","seed":202610033036,"semantic_template":"qf-factorization-identity/{\"left\":[\"-2\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-23\",\"10\",\"1\"],\"right\":[\"12\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-2, 1] and [12, 1] equal to the proposed polynomial [-23, 10, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0037","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41e16048563e1c8dfe6f9e1dcdffd6603f77cb5f12d2384bac5cb87658ab175c","family_id":"qf-factorization-identity","seed":202610033037,"semantic_template":"qf-factorization-identity/{\"left\":[\"-1\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"-13\",\"12\",\"2\"],\"right\":[\"13\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-1, 1] and [13, 1] equal to the proposed polynomial [-13, 12, 2]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0038","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fce0e86227f64dd6e7e15fb6b46a5a4721cef4d5b2ed46abf88bff46f83369ad","family_id":"qf-factorization-identity","seed":202610033038,"semantic_template":"qf-factorization-identity/{\"left\":[\"0\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"0\",\"14\",\"1\"],\"right\":[\"14\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [0, 1] and [14, 1] equal to the proposed polynomial [0, 14, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0039","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73d4e5ccc49ae02ede8125c34121abb514a4b4b76f1714d0d80d0f296eadb812","family_id":"qf-factorization-identity","seed":202610033039,"semantic_template":"qf-factorization-identity/{\"left\":[\"1\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"16\",\"16\",\"1\"],\"right\":[\"15\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [1, 1] and [15, 1] equal to the proposed polynomial [16, 16, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0040","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8f9a443e56ba2b2f2c84b534252a95809d37f614f9cf8c39cede4b63de40a6f","family_id":"qf-factorization-identity","seed":202610033040,"semantic_template":"qf-factorization-identity/{\"left\":[\"16\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"32\",\"18\",\"2\"],\"right\":[\"2\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [2, 1] and [16, 1] equal to the proposed polynomial [32, 18, 2]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0041","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"428d638698b1e5e880ac15cf9bc3c407ab1d319156023acf8825eb2467c68bdb","family_id":"qf-factorization-identity","seed":202610033041,"semantic_template":"qf-factorization-identity/{\"left\":[\"17\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"51\",\"20\",\"1\"],\"right\":[\"3\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [3, 1] and [17, 1] equal to the proposed polynomial [51, 20, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0042","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78e234569e1c122f2c916a2457bc589040a6fb50f208dbeaafa8d6d407cc47e8","family_id":"qf-factorization-identity","seed":202610033042,"semantic_template":"qf-factorization-identity/{\"left\":[\"18\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"73\",\"22\",\"1\"],\"right\":[\"4\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [4, 1] and [18, 1] equal to the proposed polynomial [73, 22, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0043","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e465774d86024190e20189f5cafcf41629764f033498ddcc7fab152279552743","family_id":"qf-factorization-identity","seed":202610033043,"semantic_template":"qf-factorization-identity/{\"left\":[\"19\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"95\",\"24\",\"2\"],\"right\":[\"5\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [5, 1] and [19, 1] equal to the proposed polynomial [95, 24, 2]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0044","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2f31c6789457866e76a3229a07fea213f2ebe3b23045188cb89a980e9596c38","family_id":"qf-factorization-identity","seed":202610033044,"semantic_template":"qf-factorization-identity/{\"left\":[\"20\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"120\",\"26\",\"1\"],\"right\":[\"6\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [6, 1] and [20, 1] equal to the proposed polynomial [120, 26, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0045","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62f453a13bb43bfef0c1e40e6db41c38d525f8238526086c2fac0c1a88df6d5e","family_id":"qf-factorization-identity","seed":202610033045,"semantic_template":"qf-factorization-identity/{\"left\":[\"21\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"148\",\"28\",\"1\"],\"right\":[\"7\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [7, 1] and [21, 1] equal to the proposed polynomial [148, 28, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0046","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09686d9a8196612a43e9938d7bccd1811c7b4a794465ac85e72bc6b6cb50b37f","family_id":"qf-factorization-identity","seed":202610033046,"semantic_template":"qf-factorization-identity/{\"left\":[\"22\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"176\",\"30\",\"2\"],\"right\":[\"8\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [8, 1] and [22, 1] equal to the proposed polynomial [176, 30, 2]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0047","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"088776f7af5b2e261829cf2c7e4ccca45d38768ede2558de441708e0fe02df57","family_id":"qf-factorization-identity","seed":202610033047,"semantic_template":"qf-factorization-identity/{\"left\":[\"23\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"207\",\"32\",\"1\"],\"right\":[\"9\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [9, 1] and [23, 1] equal to the proposed polynomial [207, 32, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0048","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"595338660761c820ed305d6622caed4bc766b2599ec0041ba7898dc3dafc1192","family_id":"qf-factorization-identity","seed":202610033048,"semantic_template":"qf-factorization-identity/{\"left\":[\"10\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"241\",\"34\",\"1\"],\"right\":[\"24\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [10, 1] and [24, 1] equal to the proposed polynomial [241, 34, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0049","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0366439ec52eb686d8afb9c8ba425935077f8132da53f43c13bc173a0ed2f762","family_id":"qf-factorization-identity","seed":202610033049,"semantic_template":"qf-factorization-identity/{\"left\":[\"11\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"275\",\"36\",\"2\"],\"right\":[\"25\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [11, 1] and [25, 1] equal to the proposed polynomial [275, 36, 2]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0050","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84bf51d9ee90bd6fa4837cd60d9e70458784429791f77ce79f208d0acb39735b","family_id":"qf-factorization-identity","seed":202610033050,"semantic_template":"qf-factorization-identity/{\"left\":[\"12\",\"1\"],\"operation\":\"identity\",\"proposed\":[\"312\",\"38\",\"1\"],\"right\":[\"26\",\"1\"]}"},"fields":{"title":"Verifying a Factorization by Coefficients","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [12, 1] and [26, 1] equal to the proposed polynomial [312, 38, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling."},"skills":["Verifying a Factorization by 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-20261003-r03-factorization-0051","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e1d12c385224add22b68293ebd9d9aaddde74db21673fe2a998cccb1c3f673b","family_id":"qf-factorization-product-coefficient","seed":202610033051,"semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"-5\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"2\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [-5, 1] and [2, -1].","answer":{"kind":"numeric","value":"-10","explanation":"Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=-10."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0052","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c330666e0ead3e7e5f5d4b47bc799a970dd9247a625fe57d8fb74979283f6e4a","family_id":"qf-factorization-product-coefficient","seed":202610033052,"semantic_template":"qf-factorization-product-coefficient/{\"k\":1,\"left\":[\"-4\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"3\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [-4, 1, 1] and [3, -1].","answer":{"kind":"numeric","value":"7","explanation":"Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0053","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fa3f97ebe8b1f61799828ab752508452f315adcd1616e47542c28083252f164","family_id":"qf-factorization-product-coefficient","seed":202610033053,"semantic_template":"qf-factorization-product-coefficient/{\"k\":2,\"left\":[\"-3\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"4\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^2 in the product of [-3, 1] and [4, -1].","answer":{"kind":"numeric","value":"-1","explanation":"Sum products of coefficients whose indices add to 2. The exact product coefficient is a_2=-1."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0054","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e88c01114e7a5b4f5dd696c6853a64042996b2d7a7517091d68229dc7cecb01","family_id":"qf-factorization-product-coefficient","seed":202610033054,"semantic_template":"qf-factorization-product-coefficient/{\"k\":3,\"left\":[\"-2\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"5\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^3 in the product of [-2, 1, 1] and [5, -1].","answer":{"kind":"numeric","value":"-1","explanation":"Sum products of coefficients whose indices add to 3. The exact product coefficient is a_3=-1."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0055","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b22ca0616de7ec6ccb80fc521794ca101764f9c4e2f558278cae926e61d35bce","family_id":"qf-factorization-product-coefficient","seed":202610033055,"semantic_template":"qf-factorization-product-coefficient/{\"k\":4,\"left\":[\"-1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"6\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^4 in the product of [-1, 1] and [6, -1].","answer":{"kind":"numeric","value":"0","explanation":"Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0056","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"830826eab9cfaac7d9176143a69b1fba108288bc39354d50b7ba873369775eb0","family_id":"qf-factorization-product-coefficient","seed":202610033056,"semantic_template":"qf-factorization-product-coefficient/{\"k\":5,\"left\":[\"0\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"7\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [0, 1, 1] and [7, -1].","answer":{"kind":"numeric","value":"0","explanation":"Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0057","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a94819923e8eb90ba6d7487f04363e84e05d3458c512e33f6992d067657befbc","family_id":"qf-factorization-product-coefficient","seed":202610033057,"semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"8\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [1, 1] and [8, -1].","answer":{"kind":"numeric","value":"8","explanation":"Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=8."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0058","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1f29b81eb8f8e618f167ff5653b6b641cd93d3745977d79e4a5639300a4c29e","family_id":"qf-factorization-product-coefficient","seed":202610033058,"semantic_template":"qf-factorization-product-coefficient/{\"k\":1,\"left\":[\"2\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"9\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [2, 1, 1] and [9, -1].","answer":{"kind":"numeric","value":"7","explanation":"Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0059","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fe36c46d0cdb61acec13039f3d897a633769ddb12d7b27d2f8896b45d784de7","family_id":"qf-factorization-product-coefficient","seed":202610033059,"semantic_template":"qf-factorization-product-coefficient/{\"k\":2,\"left\":[\"10\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"3\",\"1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^2 in the product of [3, 1] and [10, -1].","answer":{"kind":"numeric","value":"-1","explanation":"Sum products of coefficients whose indices add to 2. The exact product coefficient is a_2=-1."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0060","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9ca53514b877823549f7e149da557a496897717d02645094e352aaa91792ee9","family_id":"qf-factorization-product-coefficient","seed":202610033060,"semantic_template":"qf-factorization-product-coefficient/{\"k\":3,\"left\":[\"11\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"4\",\"1\",\"1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^3 in the product of [4, 1, 1] and [11, -1].","answer":{"kind":"numeric","value":"-1","explanation":"Sum products of coefficients whose indices add to 3. The exact product coefficient is a_3=-1."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0061","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b95f8eb060e2349ee4f393cbf11ec5a4ebbb8a02bb3f5096a84791011bb4b190","family_id":"qf-factorization-product-coefficient","seed":202610033061,"semantic_template":"qf-factorization-product-coefficient/{\"k\":4,\"left\":[\"12\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"5\",\"1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^4 in the product of [5, 1] and [12, -1].","answer":{"kind":"numeric","value":"0","explanation":"Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0062","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26f999a60071228f1fe86baa553dafb377f8154cc15e7bad331066a14afb30ae","family_id":"qf-factorization-product-coefficient","seed":202610033062,"semantic_template":"qf-factorization-product-coefficient/{\"k\":5,\"left\":[\"13\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"6\",\"1\",\"1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [6, 1, 1] and [13, -1].","answer":{"kind":"numeric","value":"0","explanation":"Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0063","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2a89cd4d638d259f52268509aa91ba651a2bbbac2801cf20b7f7c4b687d1125","family_id":"qf-factorization-product-coefficient","seed":202610033063,"semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"14\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"7\",\"1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [7, 1] and [14, -1].","answer":{"kind":"numeric","value":"98","explanation":"Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=98."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0064","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4751b883200639b5f80cbb13998702142962c32ae258bd359e7da2a5c4e7f839","family_id":"qf-factorization-product-coefficient","seed":202610033064,"semantic_template":"qf-factorization-product-coefficient/{\"k\":1,\"left\":[\"15\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"8\",\"1\",\"1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [8, 1, 1] and [15, -1].","answer":{"kind":"numeric","value":"7","explanation":"Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0065","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a206fb397b55de3b6a609463140369147acbc17695bd62dd4e9c1e251ae2ff81","family_id":"qf-factorization-product-coefficient","seed":202610033065,"semantic_template":"qf-factorization-product-coefficient/{\"k\":2,\"left\":[\"16\",\"-1\"],\"operation\":\"product_coefficient\",\"right\":[\"9\",\"1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^2 in the product of [9, 1] and [16, -1].","answer":{"kind":"numeric","value":"-1","explanation":"Sum products of coefficients whose indices add to 2. The exact product coefficient is a_2=-1."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0066","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd13f7306f8a323f2f050bfd3da6abd81098aeeb63988c39b5c17b5e8f8caed9","family_id":"qf-factorization-product-coefficient","seed":202610033066,"semantic_template":"qf-factorization-product-coefficient/{\"k\":3,\"left\":[\"10\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"17\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^3 in the product of [10, 1, 1] and [17, -1].","answer":{"kind":"numeric","value":"-1","explanation":"Sum products of coefficients whose indices add to 3. The exact product coefficient is a_3=-1."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0067","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49d0b36eef6de8f8cfc94093ecd7f33ef840c618855faaf2eda7481d506aec82","family_id":"qf-factorization-product-coefficient","seed":202610033067,"semantic_template":"qf-factorization-product-coefficient/{\"k\":4,\"left\":[\"11\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"18\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^4 in the product of [11, 1] and [18, -1].","answer":{"kind":"numeric","value":"0","explanation":"Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0068","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"138691b1948e8ebf591b15457f90b84d99357674edeff86b35a208c82b275c33","family_id":"qf-factorization-product-coefficient","seed":202610033068,"semantic_template":"qf-factorization-product-coefficient/{\"k\":5,\"left\":[\"12\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"19\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [12, 1, 1] and [19, -1].","answer":{"kind":"numeric","value":"0","explanation":"Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0069","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf93259708ee6cc9b0971d9fd76d5b23aea17af37e375342bb8d6536773a0285","family_id":"qf-factorization-product-coefficient","seed":202610033069,"semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"13\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"20\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [13, 1] and [20, -1].","answer":{"kind":"numeric","value":"260","explanation":"Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=260."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0070","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89520aad074901de6452c4f8e08a28a93d0dce6f1e96fb99d6a961909587b946","family_id":"qf-factorization-product-coefficient","seed":202610033070,"semantic_template":"qf-factorization-product-coefficient/{\"k\":1,\"left\":[\"14\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"21\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [14, 1, 1] and [21, -1].","answer":{"kind":"numeric","value":"7","explanation":"Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0071","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4b8e0669741da30fdac7a9d8272fd47810995b1ae894b4aed1246d961d138db","family_id":"qf-factorization-product-coefficient","seed":202610033071,"semantic_template":"qf-factorization-product-coefficient/{\"k\":2,\"left\":[\"15\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"22\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^2 in the product of [15, 1] and [22, -1].","answer":{"kind":"numeric","value":"-1","explanation":"Sum products of coefficients whose indices add to 2. The exact product coefficient is a_2=-1."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0072","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"544e333ba4f444fcee33ddccfe44a1f5455a4fab4da18af083c95529caf065ce","family_id":"qf-factorization-product-coefficient","seed":202610033072,"semantic_template":"qf-factorization-product-coefficient/{\"k\":3,\"left\":[\"16\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"23\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^3 in the product of [16, 1, 1] and [23, -1].","answer":{"kind":"numeric","value":"-1","explanation":"Sum products of coefficients whose indices add to 3. The exact product coefficient is a_3=-1."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0073","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aa20f921ad28381189ea4be04bbde083252d401de7eed9833a6f3bba1c57b36","family_id":"qf-factorization-product-coefficient","seed":202610033073,"semantic_template":"qf-factorization-product-coefficient/{\"k\":4,\"left\":[\"17\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"24\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^4 in the product of [17, 1] and [24, -1].","answer":{"kind":"numeric","value":"0","explanation":"Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0074","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31d95ff2a7c4eaf02ca25bf70b3344984e81bb84798ccd0f335af6f7938428b1","family_id":"qf-factorization-product-coefficient","seed":202610033074,"semantic_template":"qf-factorization-product-coefficient/{\"k\":5,\"left\":[\"18\",\"1\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"25\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [18, 1, 1] and [25, -1].","answer":{"kind":"numeric","value":"0","explanation":"Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0075","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6681d382219018f5f763e513ed64b287d68619a7d2b04fd28928d192e3392cb","family_id":"qf-factorization-product-coefficient","seed":202610033075,"semantic_template":"qf-factorization-product-coefficient/{\"k\":0,\"left\":[\"19\",\"1\"],\"operation\":\"product_coefficient\",\"right\":[\"26\",\"-1\"]}"},"fields":{"title":"Specified Coefficient of a Factor Product","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [19, 1] and [26, -1].","answer":{"kind":"numeric","value":"494","explanation":"Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=494."},"skills":["Specified Coefficient of a Factor Product"],"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-20261003-r03-factorization-0076","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"360a216d38f5d29c96a169009ce17e7ad087bccfd642cd7b76e1938902d74054","family_id":"qf-factorization-difference-squares","seed":202610033076,"semantic_template":"qf-factorization-difference-squares/{\"d\":0,\"operation\":\"difference_squares\",\"proposed\":[\"0\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(0)² equal to the proposed coefficient vector [0, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(0)² is [0, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0077","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27b9dad9f13b54a5967feab844a1551be017add797c40f7e5195a4eb514acf3c","family_id":"qf-factorization-difference-squares","seed":202610033077,"semantic_template":"qf-factorization-difference-squares/{\"d\":1,\"operation\":\"difference_squares\",\"proposed\":[\"0\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(1)² equal to the proposed coefficient vector [0, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(1)² is [-1, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0078","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80b7c004651f98aeff8ef576729e7d5efabe9ec4c72d2ed8f5f75e74dbac1772","family_id":"qf-factorization-difference-squares","seed":202610033078,"semantic_template":"qf-factorization-difference-squares/{\"d\":2,\"operation\":\"difference_squares\",\"proposed\":[\"-4\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(2)² equal to the proposed coefficient vector [-4, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(2)² is [-4, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0079","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"408b1a91245c57169d8fa78825fe3917806ba64742c502291e13e1bbc62ab8a8","family_id":"qf-factorization-difference-squares","seed":202610033079,"semantic_template":"qf-factorization-difference-squares/{\"d\":3,\"operation\":\"difference_squares\",\"proposed\":[\"-8\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(3)² equal to the proposed coefficient vector [-8, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(3)² is [-9, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0080","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd52070b3183f2b03b7485666bfef79933f3a85d1a3930e11af1af57c375088c","family_id":"qf-factorization-difference-squares","seed":202610033080,"semantic_template":"qf-factorization-difference-squares/{\"d\":4,\"operation\":\"difference_squares\",\"proposed\":[\"-16\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(4)² equal to the proposed coefficient vector [-16, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(4)² is [-16, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0081","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5ed98cb91bbeec2be81c1423bcf856367c81d7cc7c8f242671fb29f2dc7b085","family_id":"qf-factorization-difference-squares","seed":202610033081,"semantic_template":"qf-factorization-difference-squares/{\"d\":5,\"operation\":\"difference_squares\",\"proposed\":[\"-24\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(5)² equal to the proposed coefficient vector [-24, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(5)² is [-25, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0082","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae6d905f8eb30bae9d0e3bcac396c1d901ad94817938d634519df43bbed7f562","family_id":"qf-factorization-difference-squares","seed":202610033082,"semantic_template":"qf-factorization-difference-squares/{\"d\":6,\"operation\":\"difference_squares\",\"proposed\":[\"-36\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(6)² equal to the proposed coefficient vector [-36, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(6)² is [-36, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0083","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca5c72c158fa4171d20521cc012432665efa2fb0c8a57fd67242ce779bd5bbf","family_id":"qf-factorization-difference-squares","seed":202610033083,"semantic_template":"qf-factorization-difference-squares/{\"d\":7,\"operation\":\"difference_squares\",\"proposed\":[\"-48\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(7)² equal to the proposed coefficient vector [-48, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(7)² is [-49, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0084","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"937cf7439d473b8757e3fe666f5cdd760550f8e1af8f3a3c1094619807073432","family_id":"qf-factorization-difference-squares","seed":202610033084,"semantic_template":"qf-factorization-difference-squares/{\"d\":8,\"operation\":\"difference_squares\",\"proposed\":[\"-64\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(8)² equal to the proposed coefficient vector [-64, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(8)² is [-64, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0085","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48d99c94a858a987f1d62622a12a61c2c54452da5ea4f93bc1b71ea91b0be5ef","family_id":"qf-factorization-difference-squares","seed":202610033085,"semantic_template":"qf-factorization-difference-squares/{\"d\":9,\"operation\":\"difference_squares\",\"proposed\":[\"-80\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(9)² equal to the proposed coefficient vector [-80, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(9)² is [-81, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0086","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"686c3bdaa75d045239a25d14781f45ae93ec3b045a010628fb390b05b0d3533c","family_id":"qf-factorization-difference-squares","seed":202610033086,"semantic_template":"qf-factorization-difference-squares/{\"d\":10,\"operation\":\"difference_squares\",\"proposed\":[\"-100\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(10)² equal to the proposed coefficient vector [-100, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(10)² is [-100, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0087","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2216db22bbc3b8ef93ad79eb661de3f2bff7e001cfbf848ac31d795a4e3af79","family_id":"qf-factorization-difference-squares","seed":202610033087,"semantic_template":"qf-factorization-difference-squares/{\"d\":11,\"operation\":\"difference_squares\",\"proposed\":[\"-120\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(11)² equal to the proposed coefficient vector [-120, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(11)² is [-121, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0088","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c45c3cc201d4a11fe33aa2d169d04137fef3a156f9d7385c247984ebeb4dceb6","family_id":"qf-factorization-difference-squares","seed":202610033088,"semantic_template":"qf-factorization-difference-squares/{\"d\":12,\"operation\":\"difference_squares\",\"proposed\":[\"-144\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(12)² equal to the proposed coefficient vector [-144, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(12)² is [-144, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0089","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e59ab90a9d60ab43405268e7903fb29c92b781a85b1b705cde43e43b4e7104e9","family_id":"qf-factorization-difference-squares","seed":202610033089,"semantic_template":"qf-factorization-difference-squares/{\"d\":13,\"operation\":\"difference_squares\",\"proposed\":[\"-168\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(13)² equal to the proposed coefficient vector [-168, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(13)² is [-169, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0090","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e20e8be044ac2bfbf8a6fbd47696b8a57c1c542cf25ecccf5596c1063702e83","family_id":"qf-factorization-difference-squares","seed":202610033090,"semantic_template":"qf-factorization-difference-squares/{\"d\":14,\"operation\":\"difference_squares\",\"proposed\":[\"-196\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(14)² equal to the proposed coefficient vector [-196, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(14)² is [-196, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0091","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c039bd496ef8ad120c56baf802b884a9d0b9c7b56b8846f2fede084b9edca993","family_id":"qf-factorization-difference-squares","seed":202610033091,"semantic_template":"qf-factorization-difference-squares/{\"d\":15,\"operation\":\"difference_squares\",\"proposed\":[\"-224\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(15)² equal to the proposed coefficient vector [-224, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(15)² is [-225, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0092","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23a2bc6ab47a590d38e70487530708f611b0301054ef59185e2e71761787890c","family_id":"qf-factorization-difference-squares","seed":202610033092,"semantic_template":"qf-factorization-difference-squares/{\"d\":16,\"operation\":\"difference_squares\",\"proposed\":[\"-256\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(16)² equal to the proposed coefficient vector [-256, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(16)² is [-256, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0093","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"217363b9694fde6983c8f0b29b1629963c31468e8765103a7f78835ed6c0adcb","family_id":"qf-factorization-difference-squares","seed":202610033093,"semantic_template":"qf-factorization-difference-squares/{\"d\":17,\"operation\":\"difference_squares\",\"proposed\":[\"-288\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(17)² equal to the proposed coefficient vector [-288, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(17)² is [-289, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0094","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c3f435a1cb813474f39c74a6a7b367a962f34b243cd0b22a437b890491816c5","family_id":"qf-factorization-difference-squares","seed":202610033094,"semantic_template":"qf-factorization-difference-squares/{\"d\":18,\"operation\":\"difference_squares\",\"proposed\":[\"-324\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(18)² equal to the proposed coefficient vector [-324, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(18)² is [-324, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0095","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1ee2a0791f6c6e1fc79dcc31375998a729742e8641ce65909e05332f899cf73","family_id":"qf-factorization-difference-squares","seed":202610033095,"semantic_template":"qf-factorization-difference-squares/{\"d\":19,\"operation\":\"difference_squares\",\"proposed\":[\"-360\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(19)² equal to the proposed coefficient vector [-360, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(19)² is [-361, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0096","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"844f947dc87d0c24091554aae292e1742e9a023028fa34c5da0bc8e9d6232dad","family_id":"qf-factorization-difference-squares","seed":202610033096,"semantic_template":"qf-factorization-difference-squares/{\"d\":20,\"operation\":\"difference_squares\",\"proposed\":[\"-400\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(20)² equal to the proposed coefficient vector [-400, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(20)² is [-400, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0097","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a95c083dff8bd3f6885318ef4b4a1153204327821f33af2289e42647344e1c62","family_id":"qf-factorization-difference-squares","seed":202610033097,"semantic_template":"qf-factorization-difference-squares/{\"d\":21,\"operation\":\"difference_squares\",\"proposed\":[\"-440\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(21)² equal to the proposed coefficient vector [-440, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(21)² is [-441, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0098","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18cffd8bcbf6ea7606ab232b16578eb1876cc6d1f56b7f8d95f1f0fcd8152c92","family_id":"qf-factorization-difference-squares","seed":202610033098,"semantic_template":"qf-factorization-difference-squares/{\"d\":22,\"operation\":\"difference_squares\",\"proposed\":[\"-484\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(22)² equal to the proposed coefficient vector [-484, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(22)² is [-484, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0099","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbe5a0814bfd76193b8eca0756d71f76d2959bc915e430dc04af19fd4c226ac5","family_id":"qf-factorization-difference-squares","seed":202610033099,"semantic_template":"qf-factorization-difference-squares/{\"d\":23,\"operation\":\"difference_squares\",\"proposed\":[\"-528\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(23)² equal to the proposed coefficient vector [-528, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"No","explanation":"The full coefficient vector of x²−(23)² is [-529, 0, 1]. Comparing all coefficients after removing trailing zeros gives No."},"skills":["Verifying a Difference-of-Squares Identity"],"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-20261003-r03-factorization-0100","batch_id":"batch-20261003-r03-factorization-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34799889a890dff817ee9e31d569320ff32da933d17479ca950a8f441816e7f3","family_id":"qf-factorization-difference-squares","seed":202610033100,"semantic_template":"qf-factorization-difference-squares/{\"d\":24,\"operation\":\"difference_squares\",\"proposed\":[\"-576\",\"0\",\"1\"]}"},"fields":{"title":"Verifying a Difference-of-Squares Identity","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(24)² equal to the proposed coefficient vector [-576, 0, 1]? Answer Yes or No.","answer":{"kind":"short_text","value":"Yes","explanation":"The full coefficient vector of x²−(24)² is [-576, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes."},"skills":["Verifying a Difference-of-Squares Identity"],"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."}}