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R0007-04234 · q-20261008-advance-20261010-r01-polynomial-0034
Integer polynomials have ascending coefficients f=[-4, 9, -8, 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=[-4, 9, -8, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.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.
Before
f ascending coefficients=[-4, 9, -8, 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.
Rule: ascending_polynomial_identity · Uses: G001
Before
product term degrees0,0 := mul(-4, 2)
After
product term degrees0,0 = -8
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
degrees 0+0
After
contribute -8 to degree0
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S002
Before
product term degrees0,1 := mul(-4, -1)
After
product term degrees0,1 = 4
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
degrees 0+1
After
contribute 4 to degree1
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S004
Before
product term degrees0,2 := mul(-4, 1)
After
product term degrees0,2 = -4
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
degrees 0+2
After
contribute -4 to degree2
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S006
Before
product term degrees0,3 := mul(-4, 2)
After
product term degrees0,3 = -8
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
degrees 0+3
After
contribute -8 to degree3
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S008
Before
product term degrees1,0 := mul(9, 2)
After
product term degrees1,0 = 18
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
degrees 1+0
After
contribute 18 to degree1
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S010
Before
product term degrees1,1 := mul(9, -1)
After
product term degrees1,1 = -9
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
degrees 1+1
After
contribute -9 to degree2
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S012
Before
product term degrees1,2 := mul(9, 1)
After
product term degrees1,2 = 9
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
degrees 1+2
After
contribute 9 to degree3
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S014
Before
product term degrees1,3 := mul(9, 2)
After
product term degrees1,3 = 18
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
degrees 1+3
After
contribute 18 to degree4
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S016
Before
product term degrees2,0 := mul(-8, 2)
After
product term degrees2,0 = -16
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
degrees 2+0
After
contribute -16 to degree2
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S018
Before
product term degrees2,1 := mul(-8, -1)
After
product term degrees2,1 = 8
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
degrees 2+1
After
contribute 8 to degree3
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S020
Before
product term degrees2,2 := mul(-8, 1)
After
product term degrees2,2 = -8
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
degrees 2+2
After
contribute -8 to degree4
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S022
Before
product term degrees2,3 := mul(-8, 2)
After
product term degrees2,3 = -16
Conditions: none
Distributivity gives c_i d_j x^(i+j);multiply the coefficients first.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
degrees 2+3
After
contribute -16 to degree5
Conditions: none
Add degrees and assign the term to that coefficient bucket,retaining the full summed degree.
Rule: polynomial_convolution_term · Uses: G001, S024
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
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.
Rule: polynomial_convolution_term · Uses: G001, S026
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
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.
Rule: polynomial_convolution_term · Uses: G001, S028
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
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.
Rule: polynomial_convolution_term · Uses: G001, S030
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
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.
Rule: polynomial_convolution_term · Uses: G001, S032
Before
product coefficient0 := sum(-8)
After
product coefficient0 = -8
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
product coefficient1 := sum(4, 18)
After
product coefficient1 = 22
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
product coefficient2 := sum(-4, -9, -16)
After
product coefficient2 = -29
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
product coefficient3 := sum(-8, 9, 8, 2)
After
product coefficient3 = 11
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
product coefficient4 := sum(18, -8, -1)
After
product coefficient4 = 9
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
product coefficient5 := sum(-16, 1)
After
product coefficient5 = -15
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
product coefficient6 := sum(2)
After
product coefficient6 = 2
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
[-8, 22, -29, 11, 9, -15, 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].
Rule: polynomial_product_coefficients · Uses: G001, S040
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.[-8, 22, -29, 11, 9, -15, 2]
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: f091e8308f32fe307a164a4c31716144940cafc34eb658f4ebbf93dc3c56ecbacandidate record SHA-256: 3a475b5e2eeba34f51cf5ccd9df8b477091bc1ffb4136b7954edbee957a1c48fsemantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED