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R0007-04255 · q-20261008-advance-20261010-r01-polynomial-0080
Integer polynomials have ascending coefficients f=[-8, 5, -2, 1], 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=[-8, 5, -2, 1], constant term first. For c=2, 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=[-8, 5, -2, 1]
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 -8(x+(2))^0
After
expand (-8)·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.
Rule: shifted_power_binomial_identity · Uses: G001, S001
Before
(2)^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(-8, 1)
After
shift scaled binomial0,0 = -8
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
shift contribution0,0 := mul(-8, 1)
After
shift contribution0,0 = -8
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
term 5(x+(2))^1
After
expand (5)·Σ_j C(1,j)x^j(2)^(1−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, S005
Before
(2)^1
After
shift power1,0 = 2
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S006
Before
shift scaled binomial1,0 := mul(5, 1)
After
shift scaled binomial1,0 = 5
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
shift contribution1,0 := mul(5, 2)
After
shift contribution1,0 = 10
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
(2)^0
After
shift power1,1 = 1
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S009
Before
shift scaled binomial1,1 := mul(5, 1)
After
shift scaled binomial1,1 = 5
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
shift contribution1,1 := mul(5, 1)
After
shift contribution1,1 = 5
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
term -2(x+(2))^2
After
expand (-2)·Σ_j C(2,j)x^j(2)^(2−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, S012
Before
(2)^2
After
shift power2,0 = 4
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S013
Before
shift scaled binomial2,0 := mul(-2, 1)
After
shift scaled binomial2,0 = -2
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
shift contribution2,0 := mul(-2, 4)
After
shift contribution2,0 = -8
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
(2)^1
After
shift power2,1 = 2
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S016
Before
shift scaled binomial2,1 := mul(-2, 2)
After
shift scaled binomial2,1 = -4
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
shift contribution2,1 := mul(-4, 2)
After
shift contribution2,1 = -8
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
(2)^0
After
shift power2,2 = 1
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S019
Before
shift scaled binomial2,2 := mul(-2, 1)
After
shift scaled binomial2,2 = -2
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
shift contribution2,2 := mul(-2, 1)
After
shift contribution2,2 = -2
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
term 1(x+(2))^3
After
expand (1)·Σ_j C(3,j)x^j(2)^(3−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, S022
Before
(2)^3
After
shift power3,0 = 8
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S023
Before
shift scaled binomial3,0 := mul(1, 1)
After
shift scaled binomial3,0 = 1
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
shift contribution3,0 := mul(1, 8)
After
shift contribution3,0 = 8
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
(2)^2
After
shift power3,1 = 4
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S026
Before
shift scaled binomial3,1 := mul(1, 3)
After
shift scaled binomial3,1 = 3
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
shift contribution3,1 := mul(3, 4)
After
shift contribution3,1 = 12
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
(2)^1
After
shift power3,2 = 2
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S029
Before
shift scaled binomial3,2 := mul(1, 3)
After
shift scaled binomial3,2 = 3
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
shift contribution3,2 := mul(3, 2)
After
shift contribution3,2 = 6
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
(2)^0
After
shift power3,3 = 1
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S032
Before
shift scaled binomial3,3 := mul(1, 1)
After
shift scaled binomial3,3 = 1
Conditions: none
Multiply this term's coefficient by its binomial coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
shift contribution3,3 := mul(1, 1)
After
shift contribution3,3 = 1
Conditions: none
Multiply by c^(k−j) to obtain the contribution to x^j.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
shift coefficient0 := sum(-8, 10, -8, 8)
After
shift coefficient0 = 2
Conditions: none
Add all contributions to the same power.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
shift coefficient1 := sum(5, -8, 12)
After
shift coefficient1 = 9
Conditions: none
Add all contributions to the same power.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
shift coefficient2 := sum(-2, 6)
After
shift coefficient2 = 4
Conditions: none
Add all contributions to the same power.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
shift coefficient3 := sum(1)
After
shift coefficient3 = 1
Conditions: none
Add all contributions to the same power.
Rule: exact_rational_arithmetic · Uses: G001, S038
Before
[2, 9, 4, 1]
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, S039
S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S039 · premise_and_result_audit — Recompute shift coefficient3 with exact rational arithmetic, not rounded decimals.S038 · premise_and_result_audit — Recompute shift coefficient2 with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute shift coefficient1 with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute shift coefficient0 with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute shift contribution3,3 with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute shift scaled binomial3,3 with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S032 · premise_and_result_audit — Recompute shift contribution3,2 with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute shift scaled binomial3,2 with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Recompute shift contribution3,1 with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute shift scaled binomial3,1 with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Recompute shift contribution3,0 with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute shift scaled binomial3,0 with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Recompute shift contribution2,2 with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute shift scaled binomial2,2 with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Recompute shift contribution2,1 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute shift scaled binomial2,1 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Recompute shift contribution2,0 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute shift scaled binomial2,0 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Recompute shift contribution1,1 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute shift scaled binomial1,1 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute shift contribution1,0 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute shift scaled binomial1,0 with exact rational arithmetic, not rounded decimals.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.[2, 9, 4, 1]
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: 87587cb134f31d6ecc017dd3a7834fc3d03fdc21fa644e78a50e126482323aeecandidate record SHA-256: 1cb43aafa481ef283e536d692f656ccfec922cbf5856dbc5956b79504d10c6e6semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED