# Question 4226: Complete Polynomial Product Coefficients [Return to catalogue page 22](https://ocme.evemisslab.com/en/questions/page/22/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r01-polynomial-foundations/#q-20261008-advance-20261010-r01-polynomial-0026) 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. R0007-04226 · q-20261008-advance-20261010-r01-polynomial-0026 ## Original question and givens Integer polynomials have ascending coefficients f=[0] and g=[1, -1], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation. - G001 — Integer polynomials have ascending coefficients f=[0] and g=[1, -1], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation. ## 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-polynomial](https://ocme.evemisslab.com/en/mko/mko-polynomial/) ## Step-by-step derivation S001 · impliesBefore f ascending coefficients=[0] After coefficient k multiplies x^k;keep all required degrees Conditions: none Coefficients are in ascending powers with the constant at index0.Full coefficient operations are not identities inferred from one-point agreement;retain the original coefficient list. Step identifiers and dependenciesRule: ascending_polynomial_identity · Uses: G001 S002 · calculationBefore product term degrees0,0 := mul(0, 1) After product term degrees0,0 = 0 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore degrees 0+0 After contribute 0 to degree0 Conditions: none Add degrees and assign the term to that coefficient bucket,retaining the full summed degree. Step identifiers and dependenciesRule: polynomial_convolution_term · Uses: G001, S002 S004 · calculationBefore product term degrees0,1 := mul(0, -1) After product term degrees0,1 = 0 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore degrees 0+1 After contribute 0 to degree1 Conditions: none Add degrees and assign the term to that coefficient bucket,retaining the full summed degree. Step identifiers and dependenciesRule: polynomial_convolution_term · Uses: G001, S004 S006 · calculationBefore product coefficient0 := sum(0) After product coefficient0 = 0 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore product coefficient1 := sum(0) After product coefficient1 = 0 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore [0] After canonical ascending coefficient list Conditions: none The full finite convolution gives the product;two cubic factors may produce degree6,and a zero factor produces[0]. Step identifiers and dependenciesRule: polynomial_product_coefficients · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute product coefficient1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute product coefficient0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute product term degrees0,1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute product term degrees0,0 with exact rational arithmetic, not rounded decimals. - 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] 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: 010fc79f540b98e680b2b1c8fd576122b2bba2f18aced4bddd653cdf12027885 - candidate record SHA-256: c94398ffcc8ee45c804f32515856d209ff4d8bc110a118ec8be8b9649eb39631 - semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a - question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r01-polynomial-0026.json)