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R0007-04297 · q-20261008-advance-20261010-r01-polynomial-0072
Integer polynomials have ascending coefficients f=[9, 2, -1, 0], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.
G001 — Integer polynomials have ascending coefficients f=[9, 2, -1, 0], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.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=[9, 2, -1, 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.
Rule: ascending_polynomial_identity · Uses: G001
Before
[9, 2, -1, 0]
After
Horner at a=-2
Conditions: none
Use Horner from the highest nonzero degree down to the constant;do not reinterpret ascending coefficients as descending.
Rule: polynomial_horner_evaluation · Uses: G001, S001
Before
a multiply before degree1 := mul(-1, -2)
After
a multiply before degree1 = 2
Conditions: none
Multiply by the same specified input.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
a add coefficient1 := sum(2, 2)
After
a add coefficient1 = 4
Conditions: none
Then add the coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
a multiply before degree0 := mul(4, -2)
After
a multiply before degree0 = -8
Conditions: none
Multiply by the same specified input.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
a add coefficient0 := sum(-8, 9)
After
a add coefficient0 = 1
Conditions: none
Then add the coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
[9, 2, -1, 0]
After
Horner at b=2
Conditions: none
Use Horner from the highest nonzero degree down to the constant;do not reinterpret ascending coefficients as descending.
Rule: polynomial_horner_evaluation · Uses: G001, S006
Before
b multiply before degree1 := mul(-1, 2)
After
b multiply before degree1 = -2
Conditions: none
Multiply by the same specified input.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
b add coefficient1 := sum(-2, 2)
After
b add coefficient1 = 0
Conditions: none
Then add the coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
b multiply before degree0 := mul(0, 2)
After
b multiply before degree0 = 0
Conditions: none
Multiply by the same specified input.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
b add coefficient0 := sum(0, 9)
After
b add coefficient0 = 9
Conditions: none
Then add the coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
f(a)=1;f(b)=9
After
retain a,b order;two equal values do not make f constant
Conditions: none
Equal values at two points do not make a polynomial constant,as x² at−1 and1 shows;report only the specified evaluations.
Rule: two_values_not_polynomial_identity · Uses: G001, S011
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute b add coefficient0 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute b multiply before degree0 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute b add coefficient1 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute b multiply before degree1 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 — Recompute a add coefficient0 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute a multiply before degree0 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute a add coefficient1 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute a multiply before degree1 with exact rational arithmetic, not rounded decimals.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.(1, 9)
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: 32a1d2b017abb11383700cb33be763a104bbf04de2dad0811f0a1938dad0e018candidate record SHA-256: 30aba403c694d5541be303c16476c1e788c1d3d4575fa4333a325cb3c074c9cfsemantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED