# Question 4246: 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-0046) 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. R0007-04246 · q-20261008-advance-20261010-r01-polynomial-0046 ## Original question and givens Integer polynomials have ascending coefficients f=[8, 21, -20, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation. - G001 — Integer polynomials have ascending coefficients f=[8, 21, -20, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation. ## 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=[8, 21, -20, 1] 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(8, 2) After product term degrees0,0 = 16 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 16 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(8, -1) After product term degrees0,1 = -8 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 -8 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 term degrees0,2 := mul(8, 1) After product term degrees0,2 = 8 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore degrees 0+2 After contribute 8 to degree2 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, S006 S008 · calculationBefore product term degrees0,3 := mul(8, 2) After product term degrees0,3 = 16 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore degrees 0+3 After contribute 16 to degree3 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, S008 S010 · calculationBefore product term degrees1,0 := mul(21, 2) After product term degrees1,0 = 42 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore degrees 1+0 After contribute 42 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, S010 S012 · calculationBefore product term degrees1,1 := mul(21, -1) After product term degrees1,1 = -21 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore degrees 1+1 After contribute -21 to degree2 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, S012 S014 · calculationBefore product term degrees1,2 := mul(21, 1) After product term degrees1,2 = 21 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore degrees 1+2 After contribute 21 to degree3 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, S014 S016 · calculationBefore product term degrees1,3 := mul(21, 2) After product term degrees1,3 = 42 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore degrees 1+3 After contribute 42 to degree4 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, S016 S018 · calculationBefore product term degrees2,0 := mul(-20, 2) After product term degrees2,0 = -40 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore degrees 2+0 After contribute -40 to degree2 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, S018 S020 · calculationBefore product term degrees2,1 := mul(-20, -1) After product term degrees2,1 = 20 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore degrees 2+1 After contribute 20 to degree3 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, S020 S022 · calculationBefore product term degrees2,2 := mul(-20, 1) After product term degrees2,2 = -20 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore degrees 2+2 After contribute -20 to degree4 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, S022 S024 · calculationBefore product term degrees2,3 := mul(-20, 2) After product term degrees2,3 = -40 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore degrees 2+3 After contribute -40 to degree5 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, S024 S026 · calculationBefore product term degrees3,0 := mul(1, 2) After product term degrees3,0 = 2 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore degrees 3+0 After contribute 2 to degree3 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, S026 S028 · calculationBefore product term degrees3,1 := mul(1, -1) After product term degrees3,1 = -1 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore degrees 3+1 After contribute -1 to degree4 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, S028 S030 · calculationBefore product term degrees3,2 := mul(1, 1) After product term degrees3,2 = 1 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore degrees 3+2 After contribute 1 to degree5 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, S030 S032 · calculationBefore product term degrees3,3 := mul(1, 2) After product term degrees3,3 = 2 Conditions: none Distributivity gives c_i d_j x^(i+j);multiply the coefficients first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore degrees 3+3 After contribute 2 to degree6 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, S032 S034 · calculationBefore product coefficient0 := sum(16) After product coefficient0 = 16 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore product coefficient1 := sum(-8, 42) After product coefficient1 = 34 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore product coefficient2 := sum(8, -21, -40) After product coefficient2 = -53 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore product coefficient3 := sum(16, 21, 20, 2) After product coefficient3 = 59 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore product coefficient4 := sum(42, -20, -1) After product coefficient4 = 21 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore product coefficient5 := sum(-40, 1) After product coefficient5 = -39 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore product coefficient6 := sum(2) After product coefficient6 = 2 Conditions: none Sum every coefficient product with i+j=k,without truncating high powers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · impliesBefore [16, 34, -53, 59, 21, -39, 2] 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, S040 ## Reverse checks - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Recompute product coefficient6 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute product coefficient5 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute product coefficient4 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute product coefficient3 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute product coefficient2 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute product coefficient1 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute product coefficient0 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute product term degrees3,3 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Recompute product term degrees3,2 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Recompute product term degrees3,1 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute product term degrees3,0 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Recompute product term degrees2,3 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Recompute product term degrees2,2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute product term degrees2,1 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute product term degrees2,0 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute product term degrees1,3 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute product term degrees1,2 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute product term degrees1,1 with exact rational arithmetic, not rounded decimals. - 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 product term degrees1,0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute product term degrees0,3 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute product term degrees0,2 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 [16, 34, -53, 59, 21, -39, 2] 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: a3c6039e056d4b30e68b0451e73c8c09eab7d401c427718726b4148e42f4e850 - candidate record SHA-256: 99c68ef6c80c582f645f4707a46196127feb5ce6c4ae3377a799df8df95c0e74 - 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-0046.json)