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R0007-04252 · q-20261008-advance-20261010-r01-polynomial-0077
Integer polynomials have ascending coefficients f=[-4], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.
G001 — Integer polynomials have ascending coefficients f=[-4], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.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
f ascending coefficients=[-4]
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.
Rule: ascending_polynomial_identity · Uses: G001
Before
term -4(x+(-1))^0
After
expand (-4)·Σ_j C(0,j)x^j(-1)^(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.
Rule: shifted_power_binomial_identity · Uses: G001, S001
Before
(-1)^0
After
shift power0,0 = 1
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S002
Before
shift scaled binomial0,0 := mul(-4, 1)
After
shift scaled binomial0,0 = -4
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
shift contribution0,0 := mul(-4, 1)
After
shift contribution0,0 = -4
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
shift coefficient0 := sum(-4)
After
shift coefficient0 = -4
Conditions: none
Add all contributions to the same power.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
[-4]
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.
Rule: polynomial_shift_coefficients · Uses: G001, S006
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.[-4]
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: 7d3911fad617bf167ac25463c49acb5f0063a8690f17e1ee0eaa4ea6a6776ffbcandidate record SHA-256: d59b814fb40ce1b159846f1790570a803109fd2572343bf7d081d63261a09a59semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED