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R0019-10077 · q-20261003-r03-factorization-0052
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [-4, 1, 1] and [3, -1].
G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [-4, 1, 1] and [3, -1].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
Q[x];ascending coefficients;input vectors may have trailing zeros
After
equality uses every normalized coefficient,not numerical sampling
Conditions: none
Preserve the original factor order and vectors.Normalization removes only trailing high-power zeros;do not reorder source factors by semantic identity or infer polynomial equality from one point.
Rule: factorization_coefficient_identity_scope · Uses: G001
Before
raw left=[-4, 1, 1];raw right=[3, -1]
After
normalized left=[-4, 1, 1];normalized right=[3, -1]
Conditions: none
Remove trailing zeros only,not internal zeros or the original vector order.Empty and all-zero vectors both represent zero.
Rule: factor_vectors_trailing_zero_normalization · Uses: G001, S001
Before
power k=1;left=[-4, 1, 1];right=[3, -1]
After
all index pairs i+j=k=[[0, 1], [1, 0]]
Conditions: none
Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication.
Rule: product_coefficient_complete_index_pairs · Uses: G001, S002
Before
coefficient1 term0,1 := mul(-4, -1)
After
coefficient1 term0,1 = 4
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
coefficient1 term1,0 := mul(1, 3)
After
coefficient1 term1,0 = 3
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
coefficient1 complete sum := sum(4, 3)
After
coefficient1 complete sum = 7
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S005
S006 · premise_and_result_audit — Recompute coefficient1 complete sum with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute coefficient1 term1,0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute coefficient1 term0,1 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.7
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: c330666e0ead3e7e5f5d4b47bc799a970dd9247a625fe57d8fb74979283f6e4acandidate record SHA-256: 329bd0df964fd1f80ebc18d860a025a1f66c78dcdcdff62b9fc95e7541614f97semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED