# Question 10026: Verifying a Factorization by Coefficients

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0019-10026 · q-20261003-r03-factorization-0026



## Original question and givens

In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [0] and [7, 1] equal to the proposed polynomial [0]? Answer Yes or No.


- G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [0] and [7, 1] equal to the proposed polynomial [0]? Answer Yes or No.





## Definition and scope

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.


- [mko-factorization](https://ocme.evemisslab.com/en/mko/mko-factorization/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: factorization_coefficient_identity_scope · Uses: G001





S002 · impliesBefore

raw left=[0];raw right=[7, 1]

After

normalized left=[];normalized right=[7, 1]

Conditions: none

Remove trailing zeros only,not internal zeros or the original vector order.Empty and all-zero vectors both represent zero.

Step identifiers and dependenciesRule: factor_vectors_trailing_zero_normalization · Uses: G001, S001





S003 · impliesBefore

normalized factors=[[], [7, 1]]

After

zero factor gives normalized product=[0]

Conditions: none

An empty normalized vector represents the zero polynomial.A zero factor makes the entire product zero without a fabricated leading degree.

Step identifiers and dependenciesRule: zero_polynomial_product · Uses: G001, S002





S004 · impliesBefore

raw proposed=[0];normalized proposed=[0];actual product=[0]

After

coefficient identity=Yes

Conditions: none

Identity requires equality of all coefficients,including the both-zero case.Constants and trailing-zero representations do not create a different polynomial.

Step identifiers and dependenciesRule: factorization_final_full_identity_gate · Uses: G001, S003









## Reverse checks


- 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.





## Result

Yes

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 704b892bbb41ccfe5c44948731145c6e8a4a7c468900fbf4eabbc45f2fe94ef6
- candidate record SHA-256: efa7445e0ef5645ff9d64fef3f3adab9114a90dcd9dd9e24f41a5ea22adab0b2
- semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
- question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r03-factorization-0026.json)