Explore related knowledge and questions →
← Question derivation index · Original question batch
R0019-10093 · q-20261003-r03-factorization-0068
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [12, 1, 1] and [19, -1].
G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [12, 1, 1] and [19, -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=[12, 1, 1];raw right=[19, -1]
After
normalized left=[12, 1, 1];normalized right=[19, -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=5;left=[12, 1, 1];right=[19, -1]
After
all index pairs i+j=k=[]
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
power k=5;no valid contributing index pairs
After
coefficient=0
Conditions: none
With no contributing pair the empty sum is0.Do not use negative indexing or invent a higher coefficient.
Rule: empty_coefficient_sum_zero · Uses: G001, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.0
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: 138691b1948e8ebf591b15457f90b84d99357674edeff86b35a208c82b275c33candidate record SHA-256: b771fb27afa740df8d50b4ccb8611097be0f70d8cf3e10e8136d56b525343582semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED