Explore related knowledge and questions →
← Question derivation index · Original question batch
R0007-04277 · q-20261008-advance-20261010-r01-polynomial-0052
Integer polynomials have ascending coefficients f=[-3], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.
G001 — Integer polynomials have ascending coefficients f=[-3], 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=[-3]
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
[-3]
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
[-3]
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, S002
Before
f(a)=-3;f(b)=-3
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, S003
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(-3, -3)
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: 5de9e05608fc980c5bb260c0394ef4811fe40a090dee22229c32b1cea7da6ac0candidate record SHA-256: eb32021e318fa11d4a711346db1caf5cc81855d8e53f5ce65aaab0cee2448956semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED