# Question 4251: Complete Polynomial Translation 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.

R0007-04251 · q-20261008-advance-20261010-r01-polynomial-0076



## Original question and givens

Integer polynomials have ascending coefficients f=[0], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.


- G001 — Integer polynomials have ascending coefficients f=[0], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.





## 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

f ascending coefficients=[0]

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.

Step identifiers and dependenciesRule: ascending_polynomial_identity · Uses: G001





S002 · impliesBefore

term 0(x+(-2))^0

After

expand (0)·Σ_j C(0,j)x^j(-2)^(0−j)

Conditions: none

Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient.

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





S003 · impliesBefore

(-2)^0

After

shift power0,0 = 1

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

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





S004 · calculationBefore

shift scaled binomial0,0 := mul(0, 1)

After

shift scaled binomial0,0 = 0

Conditions: none

Multiply this term's coefficient by its binomial coefficient.

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





S005 · calculationBefore

shift contribution0,0 := mul(0, 1)

After

shift contribution0,0 = 0

Conditions: none

Multiply by c^(k−j) to obtain the contribution to x^j.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

shift coefficient0 := sum(0)

After

shift coefficient0 = 0

Conditions: none

Add all contributions to the same power.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

[0]

After

canonical ascending coefficient list

Conditions: none

The result is the full coefficient identity for f(x+c);constant and zero polynomials are unchanged by translation.

Step identifiers and dependenciesRule: polynomial_shift_coefficients · Uses: G001, S006









## Reverse checks


- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Recompute shift coefficient0 with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute shift contribution0,0 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute shift scaled binomial0,0 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.





## Result

[0]

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: 97244fc2ff3395ca8966381a4d3cdcd2b588124ed7b3a08e6c2e24933d947b8e
- candidate record SHA-256: 1d76c5ce54fe0b16d3305a5b80b6bf12a76a28eeed6c1fbc3dcba6da30ccbb13
- semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a
- question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r01-polynomial-0076.json)