# Question 1330: Zero-Constant Antiderivative of a Polynomial [Return to catalogue page 7](https://ocme.evemisslab.com/en/questions/page/7/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r06-antiderivative-foundations/#q-20261007-advance-20261008-r06-antiderivative-0030) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0001-01330 · q-20261007-advance-20261008-r06-antiderivative-0030 ## Original question and givens On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms. - G001 — On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-antiderivative](https://ocme.evemisslab.com/en/mko/mko-antiderivative/) ## Step-by-step derivation S001 · impliesBefore domain=R; coefficient lists are in ascending powers After constant term first; rational pairs=[numerator,denominator] Conditions: none Fix coefficient order;do not read ascending coefficients as descending or swap numerator and denominator. Step identifiers and dependenciesRule: polynomial_representation · Uses: G001 S002 · impliesBefore f coefficients ascending=[-8, 5, 0, 1] After seek F₀ with constant coefficient0 and F₀′=f Conditions: none Antiderivatives are not generally unique; this task additionally fixes the constant coefficient at0. Step identifiers and dependenciesRule: canonical_primitive_domain · Uses: G001, S001 S003 · calculationBefore primitive coefficient at degree1 := div(-8, 1) After primitive coefficient at degree1 = -8 Conditions: none Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore differentiate back coefficient0 := mul(-8, 1) After differentiate back coefficient0 = -8 Conditions: none Differentiation multiplies back by k+1 and must recover the original coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore primitive coefficient at degree2 := div(5, 2) After primitive coefficient at degree2 = 5/2 Conditions: none Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore differentiate back coefficient1 := mul(5/2, 2) After differentiate back coefficient1 = 5 Conditions: none Differentiation multiplies back by k+1 and must recover the original coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore primitive coefficient at degree3 := div(0, 3) After primitive coefficient at degree3 = 0 Conditions: none Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore differentiate back coefficient2 := mul(0, 3) After differentiate back coefficient2 = 0 Conditions: none Differentiation multiplies back by k+1 and must recover the original coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore primitive coefficient at degree4 := div(1, 4) After primitive coefficient at degree4 = 1/4 Conditions: none Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore differentiate back coefficient3 := mul(1/4, 4) After differentiate back coefficient3 = 1 Conditions: none Differentiation multiplies back by k+1 and must recover the original coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore [0, -8, 5/2, 0, 1/4] After all derivative coefficients agree with f Conditions: none Coefficient agreement proves the polynomial identity on all reals,not merely equality at a few points. Step identifiers and dependenciesRule: global_polynomial_identity · Uses: G001, S010 S012 · impliesBefore [0, -8, 5/2, 0, 1/4] After reduced coefficients;integer coefficients retain /1 Conditions: none Return ascending coefficients,trim redundant leading-in-degree zeros,and retain[0/1] for the zero polynomial. Step identifiers and dependenciesRule: coefficient_output_protocol · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute differentiate back coefficient3 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute primitive coefficient at degree4 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute differentiate back coefficient2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute primitive coefficient at degree3 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute differentiate back coefficient1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute primitive coefficient at degree2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute differentiate back coefficient0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute primitive coefficient at degree1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result [0/1, -8/1, 5/2, 0/1, 1/4] The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: a42acd29f3d34193f6c7286250c9366ae2be8fd296e67f417aad5e02fca1948a - candidate record SHA-256: c10fca61f580312d9973041a6e815715d79081c45fb1d80ac57e355d33fbe0a9 - semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc - question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r06-antiderivative-0030.json)