{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-r02-polynomial-0001","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4ea5b251a8eb6cec5e68145ee61b6f4f54f9aef643cfddc8bae30bcd7b5589f","family_id":"qf-polynomial-exact-evaluation","seed":202610032001,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[],\"operation\":\"evaluate\",\"x\":-12}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [0]. Compute p(-12) exactly.","answer":{"kind":"numeric","value":"0","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-12)=0. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0002","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32ae09c034b60edd3e4d212b48dc25428570270c4d2b1a2626ebd2ff975f8fe2","family_id":"qf-polynomial-exact-evaluation","seed":202610032002,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"5\"],\"operation\":\"evaluate\",\"x\":-11}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5]. Compute p(-11) exactly.","answer":{"kind":"numeric","value":"5","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-11)=5. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0003","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cf27c213350504ab79220d1c650e528e1b6d4b220be24a61eb0d0bdf2542d88","family_id":"qf-polynomial-exact-evaluation","seed":202610032003,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"0\",\"1\"],\"operation\":\"evaluate\",\"x\":-10}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [0, 1]. Compute p(-10) exactly.","answer":{"kind":"numeric","value":"-10","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-10)=-10. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0004","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"587d4d69e6231d2ad48a724fcbbaaedf4060bfd307249b903dbeca6b0c9ac145","family_id":"qf-polynomial-exact-evaluation","seed":202610032004,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-9\",\"0\",\"-1\"],\"operation\":\"evaluate\",\"x\":-9}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-9, 0, -1]. Compute p(-9) exactly.","answer":{"kind":"numeric","value":"-90","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-9)=-90. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0005","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab8c1fa3804a62d8f0ea35149e83342b0ee22b22999fff7ebb17511752781c8c","family_id":"qf-polynomial-exact-evaluation","seed":202610032005,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-8\",\"1\"],\"operation\":\"evaluate\",\"x\":-8}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-8, 1]. Compute p(-8) exactly.","answer":{"kind":"numeric","value":"-16","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-8)=-16. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0006","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5be7eb96430aac7c62e7e0ab328cc73608d086f3d5645b1645361632f7425eb6","family_id":"qf-polynomial-exact-evaluation","seed":202610032006,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-7\",\"2\",\"1\"],\"operation\":\"evaluate\",\"x\":-7}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-7, 2, 1]. Compute p(-7) exactly.","answer":{"kind":"numeric","value":"28","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-7)=28. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0007","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"443e58ea0a981385e85768f3f0004985a165c66885f100d14d73a203f46cd741","family_id":"qf-polynomial-exact-evaluation","seed":202610032007,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-6\",\"3\",\"-1\"],\"operation\":\"evaluate\",\"x\":-6}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-6, 3, -1]. Compute p(-6) exactly.","answer":{"kind":"numeric","value":"-60","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-6)=-60. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0008","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8778324cf76b40c184f1aeb4ea6163a9654ab2b74c7f4faa682e52f6fdc3193","family_id":"qf-polynomial-exact-evaluation","seed":202610032008,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-5\",\"-3\"],\"operation\":\"evaluate\",\"x\":-5}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-5, -3]. Compute p(-5) exactly.","answer":{"kind":"numeric","value":"10","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-5)=10. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0009","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e49c7f7865a1b595c2280ecfac0a12ad349e7a413763c150f790a85b0e9b3677","family_id":"qf-polynomial-exact-evaluation","seed":202610032009,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-4\",\"-2\",\"1\"],\"operation\":\"evaluate\",\"x\":-4}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-4, -2, 1]. Compute p(-4) exactly.","answer":{"kind":"numeric","value":"20","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-4)=20. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0010","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a72e41e8e20e0effa70f312ad451d06f1a3a9df54dba2706f399df6a188ff4fd","family_id":"qf-polynomial-exact-evaluation","seed":202610032010,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-3\",\"-1\",\"-1\"],\"operation\":\"evaluate\",\"x\":-3}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-3, -1, -1]. Compute p(-3) exactly.","answer":{"kind":"numeric","value":"-9","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-3)=-9. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0011","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73c09a12f24ef08e874cb4bf9cede89f101cac4ebe092d9c5d5b9af5604aa592","family_id":"qf-polynomial-exact-evaluation","seed":202610032011,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-2\"],\"operation\":\"evaluate\",\"x\":-2}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-2]. Compute p(-2) exactly.","answer":{"kind":"numeric","value":"-2","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-2)=-2. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0012","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b7c2aaa8a64f36e69f700a37f77052fc9ad99f126e4d8a02ea473acaa09b87c","family_id":"qf-polynomial-exact-evaluation","seed":202610032012,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"-1\",\"1\",\"1\"],\"operation\":\"evaluate\",\"x\":-1}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-1, 1, 1]. Compute p(-1) exactly.","answer":{"kind":"numeric","value":"-1","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(-1)=-1. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0013","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d5010e6db0a280637a7f38e92aa86c827241e6195782b865e83e4dde0ba1a4d","family_id":"qf-polynomial-exact-evaluation","seed":202610032013,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"0\",\"2\",\"-1\"],\"operation\":\"evaluate\",\"x\":0}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [0, 2, -1]. Compute p(0) exactly.","answer":{"kind":"numeric","value":"0","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(0)=0. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0014","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31b92e465aa9ee4f364578b6795585999f1a89efa99a7ce42c890837f22b00fb","family_id":"qf-polynomial-exact-evaluation","seed":202610032014,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"1\",\"3\"],\"operation\":\"evaluate\",\"x\":1}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [1, 3]. Compute p(1) exactly.","answer":{"kind":"numeric","value":"4","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(1)=4. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0015","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"379fd2e548d6dff316e7e45a88c063677fecb44bee661bde36c72cf1b96ae7b8","family_id":"qf-polynomial-exact-evaluation","seed":202610032015,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"2\",\"-3\",\"1\"],\"operation\":\"evaluate\",\"x\":2}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [2, -3, 1]. Compute p(2) exactly.","answer":{"kind":"numeric","value":"0","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(2)=0. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0016","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5743e26512f66d44990d9c321b1a025ad14da4d4b8aede4ac14fa4d2cb6ef17f","family_id":"qf-polynomial-exact-evaluation","seed":202610032016,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"3\",\"-2\",\"-1\"],\"operation\":\"evaluate\",\"x\":3}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [3, -2, -1]. Compute p(3) exactly.","answer":{"kind":"numeric","value":"-12","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(3)=-12. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0017","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29774781869a7edd7965ad608f5632db33d6082a03ea408b315df66bcf915c8e","family_id":"qf-polynomial-exact-evaluation","seed":202610032017,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"4\",\"-1\"],\"operation\":\"evaluate\",\"x\":4}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [4, -1]. Compute p(4) exactly.","answer":{"kind":"numeric","value":"0","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(4)=0. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0018","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86989ee086f2e1511acc84375409a8b012e767fec592b7020b1d61335140e3c7","family_id":"qf-polynomial-exact-evaluation","seed":202610032018,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"5\",\"0\",\"1\"],\"operation\":\"evaluate\",\"x\":5}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5, 0, 1]. Compute p(5) exactly.","answer":{"kind":"numeric","value":"30","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(5)=30. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0019","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"886650e0b452c09392b98c98714374f8c85123e75b09bf7acea60218c3a81574","family_id":"qf-polynomial-exact-evaluation","seed":202610032019,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"6\",\"1\",\"-1\"],\"operation\":\"evaluate\",\"x\":6}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [6, 1, -1]. Compute p(6) exactly.","answer":{"kind":"numeric","value":"-24","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(6)=-24. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0020","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13803e6e3b0beede82b697850be98b25597db6cf1d0b833fc8c1f88e0f7209d3","family_id":"qf-polynomial-exact-evaluation","seed":202610032020,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"7\",\"2\"],\"operation\":\"evaluate\",\"x\":7}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [7, 2]. Compute p(7) exactly.","answer":{"kind":"numeric","value":"21","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(7)=21. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0021","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf8fd060f773ba15e4540307eb2c138972111ea54b04f5dd3dc112d795bfad27","family_id":"qf-polynomial-exact-evaluation","seed":202610032021,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"8\",\"3\",\"1\"],\"operation\":\"evaluate\",\"x\":8}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [8, 3, 1]. Compute p(8) exactly.","answer":{"kind":"numeric","value":"96","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(8)=96. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0022","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49ee38b4a357453f0e5f8bdf10866a63da423cace2f53d9b214de60f70deee08","family_id":"qf-polynomial-exact-evaluation","seed":202610032022,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"9\",\"-3\",\"-1\"],\"operation\":\"evaluate\",\"x\":9}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [9, -3, -1]. Compute p(9) exactly.","answer":{"kind":"numeric","value":"-99","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(9)=-99. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0023","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aca6b6e2d5ca6cdfeb78195a0ae0be445e4a50e0e04f31b70a8707f858bba37e","family_id":"qf-polynomial-exact-evaluation","seed":202610032023,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"10\",\"-2\"],\"operation\":\"evaluate\",\"x\":10}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [10, -2]. Compute p(10) exactly.","answer":{"kind":"numeric","value":"-10","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(10)=-10. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0024","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2630b5b71e8f8bcbad648ee2bc65c277cb7d156c53c80d843c0680c029138439","family_id":"qf-polynomial-exact-evaluation","seed":202610032024,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"11\",\"-1\",\"1\"],\"operation\":\"evaluate\",\"x\":11}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [11, -1, 1]. Compute p(11) exactly.","answer":{"kind":"numeric","value":"121","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(11)=121. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0025","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a99f4c4ac0d8ac90e5baca271a2ff9ef4c4dcd168940a1936b57413cc0c6fed","family_id":"qf-polynomial-exact-evaluation","seed":202610032025,"semantic_template":"qf-polynomial-exact-evaluation/{\"coefficients\":[\"12\",\"0\",\"-1\"],\"operation\":\"evaluate\",\"x\":12}"},"fields":{"title":"Exact Polynomial Evaluation","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [12, 0, -1]. Compute p(12) exactly.","answer":{"kind":"numeric","value":"-132","explanation":"Multiply each coefficient by the corresponding natural-number power and add: p(12)=-132. This is evaluation at one input and does not establish a general polynomial identity."},"skills":["Exact Polynomial Evaluation"],"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-r02-polynomial-0026","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8905a1a0c9e252d5461acfd00e95bd863463c4f3c86bc587c81bd7509f9da880","family_id":"qf-polynomial-coefficient","seed":202610032026,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[],\"k\":0,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [0], find a_0, the coefficient of x^0.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_0=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0027","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b274d4772f4cc4fce234ffa7f7170f437359929bbd55a6eb5e4302f1a422402","family_id":"qf-polynomial-coefficient","seed":202610032027,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-9\",\"-1\",\"2\"],\"k\":1,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-9, -1, 2], find a_1, the coefficient of x^1.","answer":{"kind":"numeric","value":"-1","explanation":"The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0028","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d017c472b7a2d37a2df395193f7a4c168c339d03649a1a1243c775b6a793b9a","family_id":"qf-polynomial-coefficient","seed":202610032028,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-8\",\"0\",\"3\"],\"k\":2,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-8, 0, 3], find a_2, the coefficient of x^2.","answer":{"kind":"numeric","value":"3","explanation":"The index is the nonnegative exponent, so a_2=3. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0029","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd926281715015e3ee6c9c32a3a23cc1c9273d0e201a36dd20df24c90650ce73","family_id":"qf-polynomial-coefficient","seed":202610032029,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-7\",\"1\",\"4\"],\"k\":3,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-7, 1, 4], find a_3, the coefficient of x^3.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0030","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6a5d402df6021189f3fc1a7a980ef9f8cfc8315775e7a9dfc78e08221d8800d","family_id":"qf-polynomial-coefficient","seed":202610032030,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-6\",\"2\",\"1\"],\"k\":4,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-6, 2, 1], find a_4, the coefficient of x^4.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0031","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e07d94984082b0739baac60b5b460d2d166ae3b15fbf8c184f0f328338d7fbc","family_id":"qf-polynomial-coefficient","seed":202610032031,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-5\",\"-2\",\"2\"],\"k\":0,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-5, -2, 2], find a_0, the coefficient of x^0.","answer":{"kind":"numeric","value":"-5","explanation":"The index is the nonnegative exponent, so a_0=-5. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0032","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdad6ef9cc30ceb72d82ce41b41ae48c3d8bbade2e4139c57d0299f06a48c8b2","family_id":"qf-polynomial-coefficient","seed":202610032032,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-4\",\"-1\",\"3\"],\"k\":1,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-4, -1, 3], find a_1, the coefficient of x^1.","answer":{"kind":"numeric","value":"-1","explanation":"The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0033","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a722cdeae3e780f32e11ee43893ae640a64c227547b2ef7ef234a19a6ec1214","family_id":"qf-polynomial-coefficient","seed":202610032033,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-3\",\"0\",\"4\"],\"k\":2,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-3, 0, 4], find a_2, the coefficient of x^2.","answer":{"kind":"numeric","value":"4","explanation":"The index is the nonnegative exponent, so a_2=4. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0034","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09ec6da87f695c227fef56696a4eb996355e2cc3610a9fc5558f341e7fde72d5","family_id":"qf-polynomial-coefficient","seed":202610032034,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-2\",\"1\",\"1\"],\"k\":3,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-2, 1, 1], find a_3, the coefficient of x^3.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0035","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d5c50083b032e49343aa829205a204ed6f7350b98d19e2123773b8c372cb71e","family_id":"qf-polynomial-coefficient","seed":202610032035,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"-1\",\"2\",\"2\"],\"k\":4,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [-1, 2, 2], find a_4, the coefficient of x^4.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0036","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8263c8f5bb2ece5ce9f888535d7ebdcd3e68af5353b2e9bb55c3396b90774e25","family_id":"qf-polynomial-coefficient","seed":202610032036,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"0\",\"-2\",\"3\"],\"k\":0,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [0, -2, 3], find a_0, the coefficient of x^0.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_0=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0037","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69b71088be36e3e88ec0aefc62eeadfd432d76fcf3b7605b83168944c59f74e1","family_id":"qf-polynomial-coefficient","seed":202610032037,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"1\",\"-1\",\"4\"],\"k\":1,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [1, -1, 4], find a_1, the coefficient of x^1.","answer":{"kind":"numeric","value":"-1","explanation":"The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0038","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00cbd47bcedec6577b8f6ba5f479d529004154d32a601ae7c6e04338c13785ed","family_id":"qf-polynomial-coefficient","seed":202610032038,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"2\",\"0\",\"1\"],\"k\":2,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [2, 0, 1], find a_2, the coefficient of x^2.","answer":{"kind":"numeric","value":"1","explanation":"The index is the nonnegative exponent, so a_2=1. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0039","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb479cadeca22560fd48a07c19a4e551db31f9dbdea98830e0acb429d6ef603d","family_id":"qf-polynomial-coefficient","seed":202610032039,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"3\",\"1\",\"2\"],\"k\":3,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [3, 1, 2], find a_3, the coefficient of x^3.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0040","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c35689fb06b2c5cdee9f6dabd99f4181371f1672b3b6d8ffa74de55c83ee43c","family_id":"qf-polynomial-coefficient","seed":202610032040,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"4\",\"2\",\"3\"],\"k\":4,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [4, 2, 3], find a_4, the coefficient of x^4.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0041","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4656a424cc2bd3070527d6f95a7d7bbd8cbb74f999d66036031d1f2ba136792","family_id":"qf-polynomial-coefficient","seed":202610032041,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"5\",\"-2\",\"4\"],\"k\":0,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [5, -2, 4], find a_0, the coefficient of x^0.","answer":{"kind":"numeric","value":"5","explanation":"The index is the nonnegative exponent, so a_0=5. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0042","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc80f8f56b3e835149155653847ff8f53b10334958ed784182857ca2ab689ad4","family_id":"qf-polynomial-coefficient","seed":202610032042,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"6\",\"-1\",\"1\"],\"k\":1,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [6, -1, 1], find a_1, the coefficient of x^1.","answer":{"kind":"numeric","value":"-1","explanation":"The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0043","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71764d8e6bd62104a3ce698f98e352660bf8c0be30ae868edd8d85e602818de1","family_id":"qf-polynomial-coefficient","seed":202610032043,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"7\",\"0\",\"2\"],\"k\":2,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [7, 0, 2], find a_2, the coefficient of x^2.","answer":{"kind":"numeric","value":"2","explanation":"The index is the nonnegative exponent, so a_2=2. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0044","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86f2c7d5e1ed3051636a58096d3ca582a5204152c7bb02935f4f959a244a1955","family_id":"qf-polynomial-coefficient","seed":202610032044,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"8\",\"1\",\"3\"],\"k\":3,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [8, 1, 3], find a_3, the coefficient of x^3.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0045","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eac2f56ccfbfa73c716dd6ee01048cb7e63ff2ee008c39ca9aa65fa3dc3f0a2","family_id":"qf-polynomial-coefficient","seed":202610032045,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"9\",\"2\",\"4\"],\"k\":4,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [9, 2, 4], find a_4, the coefficient of x^4.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0046","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6f60362fb0b0340cd0ed8e73bcf2bcf463821173170b18bb489a6c1337e5a48","family_id":"qf-polynomial-coefficient","seed":202610032046,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"10\",\"-2\",\"1\"],\"k\":0,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [10, -2, 1], find a_0, the coefficient of x^0.","answer":{"kind":"numeric","value":"10","explanation":"The index is the nonnegative exponent, so a_0=10. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0047","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"835ea11b7a91d12df32d125d184a4dff6163da5c49d51178ffbca4f18cc5bd1f","family_id":"qf-polynomial-coefficient","seed":202610032047,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"11\",\"-1\",\"2\"],\"k\":1,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [11, -1, 2], find a_1, the coefficient of x^1.","answer":{"kind":"numeric","value":"-1","explanation":"The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0048","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db3742219555d01d8b8d98d8232457ae25b091734d912be43d2337bc0404134d","family_id":"qf-polynomial-coefficient","seed":202610032048,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"12\",\"0\",\"3\"],\"k\":2,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [12, 0, 3], find a_2, the coefficient of x^2.","answer":{"kind":"numeric","value":"3","explanation":"The index is the nonnegative exponent, so a_2=3. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0049","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c09e805d241d0a5c883a9ff1fd6450fbef5985d911101156131dc51fd5ad76b","family_id":"qf-polynomial-coefficient","seed":202610032049,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"13\",\"1\",\"4\"],\"k\":3,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [13, 1, 4], find a_3, the coefficient of x^3.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0050","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26ce39d09e94c56feefb659d0a0a709d05472b2bf75b3d1e96327163ff878d05","family_id":"qf-polynomial-coefficient","seed":202610032050,"semantic_template":"qf-polynomial-coefficient/{\"coefficients\":[\"14\",\"2\",\"1\"],\"k\":4,\"operation\":\"coefficient\"}"},"fields":{"title":"Coefficient of a Specified Power","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. For [14, 2, 1], find a_4, the coefficient of x^4.","answer":{"kind":"numeric","value":"0","explanation":"The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero."},"skills":["Coefficient of a Specified Power"],"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-r02-polynomial-0051","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0feedd6eb84062f66a85169d40ac9683adfbcb0a2487ca5fe09c118692bec487","family_id":"qf-polynomial-degree","seed":202610032051,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [0]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"undefined","explanation":"All coefficients are zero, so this entry calls the degree undefined. Trailing zeros do not change the polynomial."},"skills":["Polynomial Degree"],"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-r02-polynomial-0052","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f560678f57efe685e391d31dbfa17e7f6ec8a46b4c0a9f9239d3633b0ec0def","family_id":"qf-polynomial-degree","seed":202610032052,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-4\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-4]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"0","explanation":"The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0053","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f55a5d2ae172d60f928fe763187322098956c8f8823da130bf7f64fa4f204146","family_id":"qf-polynomial-degree","seed":202610032053,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-10\",\"0\",\"1\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-10, 0, 1]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"2","explanation":"The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0054","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4afcf398c6a602ec349ca5e700b688958014fb06906b131885148123f00afd67","family_id":"qf-polynomial-degree","seed":202610032054,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-9\",\"0\",\"0\",\"1\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-9, 0, 0, 1]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"3","explanation":"The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0055","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d27e402c172e63b70e4092e5656266569a64d25949df0e46170685cb565ec6d","family_id":"qf-polynomial-degree","seed":202610032055,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-8\",\"0\",\"0\",\"0\",\"1\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-8, 0, 0, 0, 1]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"4","explanation":"The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0056","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c47f8e99639e54df86882a6ec2dce59c87a2a3423d495e67a57e51bba870cd07","family_id":"qf-polynomial-degree","seed":202610032056,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"2\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [2]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"0","explanation":"The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0057","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"046ed81fa322a37d84e50a451c59fb08529649c41582d0060e52faeb85fc6f98","family_id":"qf-polynomial-degree","seed":202610032057,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-6\",\"2\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-6, 2]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"1","explanation":"The largest index with a nonzero coefficient is 1, so the degree is 1. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0058","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89c64436edcc072260695db3866ac525883e668d96feef96d60764d4026e6967","family_id":"qf-polynomial-degree","seed":202610032058,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-5\",\"0\",\"2\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-5, 0, 2]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"2","explanation":"The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0059","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56db099bead56a40634860195b26c13a833ffbbd09245982894c6cf07eaec266","family_id":"qf-polynomial-degree","seed":202610032059,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-4\",\"0\",\"0\",\"2\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-4, 0, 0, 2]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"3","explanation":"The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0060","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57f0ef7664286fe334c489bccbb732fbf4a9ff17b50be9215ee1e48851c4420c","family_id":"qf-polynomial-degree","seed":202610032060,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-3\",\"0\",\"0\",\"0\",\"2\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-3, 0, 0, 0, 2]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"4","explanation":"The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0061","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60978e2336d7eabcb577bc4b4ae601ccc3017dc39792a7a5f0d69c1ffe4092b4","family_id":"qf-polynomial-degree","seed":202610032061,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"3\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [3]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"0","explanation":"The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0062","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"598358917eec85f810a334a0b4cc0698d023a3ff99c33b301224faa14777f176","family_id":"qf-polynomial-degree","seed":202610032062,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"-1\",\"3\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-1, 3]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"1","explanation":"The largest index with a nonzero coefficient is 1, so the degree is 1. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0063","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2b2f03f3813c179b8c4ced2540d46213080da0a1de40dd8566ab5202249551e","family_id":"qf-polynomial-degree","seed":202610032063,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"0\",\"0\",\"3\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [0, 0, 3]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"2","explanation":"The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0064","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca4bea68df4efd2731a3563e473a323c4e84fdbd9711595aafd5f335af039685","family_id":"qf-polynomial-degree","seed":202610032064,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"1\",\"0\",\"0\",\"3\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [1, 0, 0, 3]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"3","explanation":"The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0065","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d534677420800a4b262023904a261ddc8e2c4051d61837b13597dcb51b09959b","family_id":"qf-polynomial-degree","seed":202610032065,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"2\",\"0\",\"0\",\"0\",\"3\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [2, 0, 0, 0, 3]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"4","explanation":"The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0066","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9eee5724f9cd81d73b769e06c46af476173a54c0650bccccc7856f44b8a0b56","family_id":"qf-polynomial-degree","seed":202610032066,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"4\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [4]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"0","explanation":"The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0067","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cffdbec752ce22a43a5d4f3102b3d1bcdf26f7ea0579ba32540c43d21fbc3f4","family_id":"qf-polynomial-degree","seed":202610032067,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"4\",\"4\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [4, 4]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"1","explanation":"The largest index with a nonzero coefficient is 1, so the degree is 1. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0068","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a23909b8e7b066145104a60255959019828b25e1a58ace4f625ddcf7888f93c6","family_id":"qf-polynomial-degree","seed":202610032068,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"5\",\"0\",\"4\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [5, 0, 4]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"2","explanation":"The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0069","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cae71b8e3d1cc16ee9d118681adbc5d2b28b11a98104fdf2ba79f7383a9585d","family_id":"qf-polynomial-degree","seed":202610032069,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"6\",\"0\",\"0\",\"4\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [6, 0, 0, 4]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"3","explanation":"The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0070","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cda8dd4e64bbfd0d378f5952f2a6e91e461693251cf1edd0e24fc7d12949a9ec","family_id":"qf-polynomial-degree","seed":202610032070,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"7\",\"0\",\"0\",\"0\",\"4\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [7, 0, 0, 0, 4]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"4","explanation":"The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0071","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a44b9b8950416ac3f7361cd07d9ecb247eca2a3c6e4c5cbd541780c7d78f7a05","family_id":"qf-polynomial-degree","seed":202610032071,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"5\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [5]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"0","explanation":"The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0072","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d775b4a482c8ae94a3ee5612df3b24a9bd6aa14e33328104351ce692381b9458","family_id":"qf-polynomial-degree","seed":202610032072,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"9\",\"5\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [9, 5]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"1","explanation":"The largest index with a nonzero coefficient is 1, so the degree is 1. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0073","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"562f148c43b6f0d398305448d39f4f0fa4e2385994d19c0578349ee8ecbd88c8","family_id":"qf-polynomial-degree","seed":202610032073,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"10\",\"0\",\"5\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [10, 0, 5]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"2","explanation":"The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0074","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c12ebd6cb6cfebcc4f9af0b4ce6698006a0bd2c2e9163a74707d877fd7433e94","family_id":"qf-polynomial-degree","seed":202610032074,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"11\",\"0\",\"0\",\"5\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [11, 0, 0, 5]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"3","explanation":"The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0075","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"beb3da6b52679a52812d0a204cf9eb2fda85792d96dd2c64a6304708ee0112a2","family_id":"qf-polynomial-degree","seed":202610032075,"semantic_template":"qf-polynomial-degree/{\"coefficients\":[\"12\",\"0\",\"0\",\"0\",\"5\"],\"operation\":\"degree\"}"},"fields":{"title":"Polynomial Degree","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [12, 0, 0, 0, 5]. Answer undefined for the zero polynomial.","answer":{"kind":"short_text","value":"4","explanation":"The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it."},"skills":["Polynomial Degree"],"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-r02-polynomial-0076","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c6889fbca07bb54baf8a104eb986474c5af1ad8375af1900d80e25e95fd7d3d","family_id":"qf-polynomial-addition","seed":202610032076,"semantic_template":"qf-polynomial-addition/{\"left\":[],\"operation\":\"add\",\"right\":[]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [0] and [0]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0077","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86d84133723489e8d771036e8ba48dba645b967a0b9b37ee6349047efa0ea700","family_id":"qf-polynomial-addition","seed":202610032077,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-2\",\"-3\"],\"operation\":\"add\",\"right\":[\"2\",\"3\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [2, 3] and [-2, -3]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0078","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5a6f51c10ac3bee92cebefeab152c2f37414017ac0266f7c46fd6db2cee991e","family_id":"qf-polynomial-addition","seed":202610032078,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-10\",\"3\"],\"operation\":\"add\",\"right\":[\"10\",\"1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-10, 3] and [10, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 4]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 4]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0079","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90f764b28068e339ac40a94c37d82235d44b85db7b3c2e1f004acf64c4695533","family_id":"qf-polynomial-addition","seed":202610032079,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-9\",\"4\"],\"operation\":\"add\",\"right\":[\"9\",\"-1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-9, 4] and [9, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 3]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0080","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24764fc2213e4a82228cf9581130d5cd97fd9eececf625645c3e368d39583acf","family_id":"qf-polynomial-addition","seed":202610032080,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-8\",\"1\"],\"operation\":\"add\",\"right\":[\"8\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-8, 1] and [8]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 1]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 1]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0081","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2cb49056c0ba3a2d420c943b6657cbdc99a0840fcad10a20ff198c7d8834315","family_id":"qf-polynomial-addition","seed":202610032081,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-7\",\"2\"],\"operation\":\"add\",\"right\":[\"7\",\"1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-7, 2] and [7, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 3]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0082","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37d91df14601e9cc16c1f8d185cb376419104ca0c68d7f8307cc978d84984523","family_id":"qf-polynomial-addition","seed":202610032082,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-6\",\"3\"],\"operation\":\"add\",\"right\":[\"6\",\"-1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-6, 3] and [6, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 2]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0083","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bdb420e1a0ef322ee717fa666411a81f9f40845b6ffdea316d4a213b3df2b91","family_id":"qf-polynomial-addition","seed":202610032083,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-5\",\"4\"],\"operation\":\"add\",\"right\":[\"5\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-5, 4] and [5]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 4]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 4]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0084","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23dcf59c2e66f7d72ec513b3eb8a0888b461f0f2812590477ddda28286672b63","family_id":"qf-polynomial-addition","seed":202610032084,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-4\",\"1\"],\"operation\":\"add\",\"right\":[\"4\",\"1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-4, 1] and [4, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 2]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0085","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98a7e6e89fbb36ed50f572c93d7a36889e15c28b073d775d1ee82a9b01adff79","family_id":"qf-polynomial-addition","seed":202610032085,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-3\",\"2\"],\"operation\":\"add\",\"right\":[\"3\",\"-1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-3, 2] and [3, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 1]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 1]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0086","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2757dd34f1c26c35aa36656d54cac9e2e5a585e2f7f5e0f37a36b3a081a7f9b8","family_id":"qf-polynomial-addition","seed":202610032086,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-2\",\"3\"],\"operation\":\"add\",\"right\":[\"2\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-2, 3] and [2]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 3]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0087","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3613dd0c83fe4063fd5f319c70ec9be1793260d9ca3a9d6c2e3a13fa2991a263","family_id":"qf-polynomial-addition","seed":202610032087,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-1\",\"4\"],\"operation\":\"add\",\"right\":[\"1\",\"1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-1, 4] and [1, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 5]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 5]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0088","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a930625e218868698c9f4fcb7b20eab45f682f93a4f47980bd7a5236e7a7848","family_id":"qf-polynomial-addition","seed":202610032088,"semantic_template":"qf-polynomial-addition/{\"left\":[\"0\",\"-1\"],\"operation\":\"add\",\"right\":[\"0\",\"1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [0, 1] and [0, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0089","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97dbf2ecfeaf1b1311e4af8aeb08ba5d11d51d5144ff20e00a362fd76348882b","family_id":"qf-polynomial-addition","seed":202610032089,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-1\"],\"operation\":\"add\",\"right\":[\"1\",\"2\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [1, 2] and [-1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 2]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0090","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa20ab58c759f682699b21248fda551d100b1165d69d47f6f8f61fe19bbc6921","family_id":"qf-polynomial-addition","seed":202610032090,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-2\",\"1\"],\"operation\":\"add\",\"right\":[\"2\",\"3\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [2, 3] and [-2, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 4]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 4]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0091","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c01aaaf45cb91aedc556690267f51539f9df59927e872e5f234db3944e2a10c","family_id":"qf-polynomial-addition","seed":202610032091,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-3\",\"-1\"],\"operation\":\"add\",\"right\":[\"3\",\"4\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [3, 4] and [-3, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 3]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0092","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80fe7c23d952173d6eaf66c9dd17ce9cab4b8f5a0ecb4d2bb39be9e2f068982d","family_id":"qf-polynomial-addition","seed":202610032092,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-4\"],\"operation\":\"add\",\"right\":[\"4\",\"1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [4, 1] and [-4]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 1]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 1]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0093","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42cc0aacedd3df0620cf4c11d36a27b1ad0afcbf813c24e7717e335826f1674c","family_id":"qf-polynomial-addition","seed":202610032093,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-5\",\"1\"],\"operation\":\"add\",\"right\":[\"5\",\"2\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [5, 2] and [-5, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 3]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0094","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38be57dbe6a3acb937a5ffa8661f1ee8192ebd905604610d4723434a597575fc","family_id":"qf-polynomial-addition","seed":202610032094,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-6\",\"-1\"],\"operation\":\"add\",\"right\":[\"6\",\"3\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [6, 3] and [-6, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 2]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0095","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b03f5538395e823158cadf118b09b79f7947e14148b7cb383bfbef6a28734ee8","family_id":"qf-polynomial-addition","seed":202610032095,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-7\"],\"operation\":\"add\",\"right\":[\"7\",\"4\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [7, 4] and [-7]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 4]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 4]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0096","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38d2168b3429852b17657a5468e53858955cbe6a5e7cd8516da176eb2ce8f518","family_id":"qf-polynomial-addition","seed":202610032096,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-8\",\"1\"],\"operation\":\"add\",\"right\":[\"8\",\"1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [8, 1] and [-8, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 2]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0097","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04bfd5a475fbf3c5540f810ec1c48a79a560f9797694e044ce3fbf2af88c5b33","family_id":"qf-polynomial-addition","seed":202610032097,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-9\",\"-1\"],\"operation\":\"add\",\"right\":[\"9\",\"2\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [9, 2] and [-9, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 1]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 1]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0098","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"622a23eb7ec6801255e19a66b201805ccabedec93699e3613ac3103e49d49287","family_id":"qf-polynomial-addition","seed":202610032098,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-10\"],\"operation\":\"add\",\"right\":[\"10\",\"3\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [10, 3] and [-10]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 3]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0099","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96994b689ef1c8ea6499709c6cf3f3381a58637c9dbb8874bc6237c5560322d5","family_id":"qf-polynomial-addition","seed":202610032099,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-11\",\"1\"],\"operation\":\"add\",\"right\":[\"11\",\"4\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [11, 4] and [-11, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0, 5]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0, 5]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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-r02-polynomial-0100","batch_id":"batch-20261003-r02-polynomial-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a87ba7f8fab64e57b5a11baef6444c18d49342b7392d0957fd90357d6fd4d193","family_id":"qf-polynomial-addition","seed":202610032100,"semantic_template":"qf-polynomial-addition/{\"left\":[\"-12\",\"-1\"],\"operation\":\"add\",\"right\":[\"12\",\"1\"]}"},"fields":{"title":"Coefficientwise Polynomial Addition","stem":"In Q[x], coefficient vectors are ordered from lowest to highest power. Add [12, 1] and [-12, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].","answer":{"kind":"expression","value":"[0]","explanation":"Add coefficients of each equal power and remove trailing zeros. The sum is [0]; the zero polynomial is represented by [0]."},"skills":["Coefficientwise Polynomial Addition"],"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."}}