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R0007-04223 · q-20261008-advance-20261010-r01-polynomial-0023
Integer polynomials have ascending coefficients f=[10, 23, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.
G001 — Integer polynomials have ascending coefficients f=[10, 23, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.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=[10, 23, 0, 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
sum coefficient0 := sum(10, 1)
After
sum coefficient0 = 11
Conditions: none
Combine coefficients of the same power,padding a missing term with0.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
sum coefficient1 := sum(23, -2)
After
sum coefficient1 = 21
Conditions: none
Combine coefficients of the same power,padding a missing term with0.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
sum coefficient2 := sum(0, 2)
After
sum coefficient2 = 2
Conditions: none
Combine coefficients of the same power,padding a missing term with0.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
sum coefficient3 := sum(1, -1)
After
sum coefficient3 = 0
Conditions: none
Combine coefficients of the same power,padding a missing term with0.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
[11, 21, 2]
After
canonical ascending coefficient list
Conditions: none
Coefficientwise addition preserves the polynomial identity;remove trailing zeros but retain[0] for the zero polynomial.
Rule: polynomial_sum_coefficients · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute sum coefficient3 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute sum coefficient2 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute sum coefficient1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute sum coefficient0 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.[11, 21, 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: 2abc43e73afdc74e0b61e881ec1f192f11a449146066253332d2461d131cdbd7candidate record SHA-256: f3448dda9d73d16f97251f5a05257c43b2117c93da2d11d95cf07f32b13b8b03semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED