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R0007-04227 · q-20261008-advance-20261010-r01-polynomial-0027
Integer polynomials have ascending coefficients f=[2] and g=[-3], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.
G001 — Integer polynomials have ascending coefficients f=[2] and g=[-3], 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=[2]
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(2, -3)
After
product term degrees0,0 = -6
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 -6 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 coefficient0 := sum(-6)
After
product coefficient0 = -6
Conditions: none
Sum every coefficient product with i+j=k,without truncating high powers.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
[-6]
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, S004
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 coefficient0 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.[-6]
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.
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