# Question 34231: Coefficient of a Specified Power

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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.

R0067-34231 · q-20261003-r02-polynomial-0031



## Original question and givens

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-5, -2, 2], find a_0, the coefficient of x^0.


- G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. For [-5, -2, 2], find a_0, the coefficient of x^0.





## 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-polynomial](https://ocme.evemisslab.com/en/mko/mko-polynomial/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-polynomial-coefficient, [coefficients, k, operation]]

After

derived = [coefficient, no-invented-domain]

Conditions: none

Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.

Step identifiers and dependenciesRule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001





S002 · impliesBefore

source roles = [polynomial ORIGINAL, [-5, -2, 2]]

After

derived = [[-5, -2, 2], [-5, -2, 2]]

Conditions: none

In Q[x],coefficients run from low to high degree.Type EVERY RAW SAFE cell of a0..32-entry list,including trailing zeros.Trim ONLY trailing zeros,not internal zeros.The zero polynomial has empty internal support but displays[0];keep the ORIGINAL source vector unchanged.

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





S003 · impliesBefore

source roles = [[-5, -2, 2], 0]

After

derived = [-5]

Conditions: none

Index k is the NONNEGATIVE exponent of x^k,not a one-based list position.Valid k0..64 beyond finite support has coefficient0.Do not delete internal zeros and renumber powers.

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









## Reverse checks


- S003 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
- S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
- S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.





## Result

-5

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: 7e07d94984082b0739baac60b5b460d2d166ae3b15fbf8c184f0f328338d7fbc
- candidate record SHA-256: 5b386ce5cae081038ece60441bab2cac451a09507baba4c7c89c2812a452748c
- semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
- question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r02-polynomial-0031.json)