# Question 4278: Ordered Polynomial Values at Two Points [Return to catalogue page 22](https://ocme.evemisslab.com/en/questions/page/22/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r01-polynomial-foundations/#q-20261008-advance-20261010-r01-polynomial-0053) 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. R0007-04278 · q-20261008-advance-20261010-r01-polynomial-0053 ## Original question and givens Integer polynomials have ascending coefficients f=[-10, 2, -1, 2], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers. - G001 — Integer polynomials have ascending coefficients f=[-10, 2, -1, 2], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers. ## 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=[-10, 2, -1, 2] 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 [-10, 2, -1, 2] 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. Step identifiers and dependenciesRule: polynomial_horner_evaluation · Uses: G001, S001 S003 · calculationBefore a multiply before degree2 := mul(2, -2) After a multiply before degree2 = -4 Conditions: none Multiply by the same specified input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore a add coefficient2 := sum(-4, -1) After a add coefficient2 = -5 Conditions: none Then add the coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore a multiply before degree1 := mul(-5, -2) After a multiply before degree1 = 10 Conditions: none Multiply by the same specified input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore a add coefficient1 := sum(10, 2) After a add coefficient1 = 12 Conditions: none Then add the coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore a multiply before degree0 := mul(12, -2) After a multiply before degree0 = -24 Conditions: none Multiply by the same specified input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore a add coefficient0 := sum(-24, -10) After a add coefficient0 = -34 Conditions: none Then add the coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore [-10, 2, -1, 2] After Horner at b=3 Conditions: none Use Horner from the highest nonzero degree down to the constant;do not reinterpret ascending coefficients as descending. Step identifiers and dependenciesRule: polynomial_horner_evaluation · Uses: G001, S008 S010 · calculationBefore b multiply before degree2 := mul(2, 3) After b multiply before degree2 = 6 Conditions: none Multiply by the same specified input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore b add coefficient2 := sum(6, -1) After b add coefficient2 = 5 Conditions: none Then add the coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore b multiply before degree1 := mul(5, 3) After b multiply before degree1 = 15 Conditions: none Multiply by the same specified input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore b add coefficient1 := sum(15, 2) After b add coefficient1 = 17 Conditions: none Then add the coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore b multiply before degree0 := mul(17, 3) After b multiply before degree0 = 51 Conditions: none Multiply by the same specified input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore b add coefficient0 := sum(51, -10) After b add coefficient0 = 41 Conditions: none Then add the coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore f(a)=-34;f(b)=41 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. Step identifiers and dependenciesRule: two_values_not_polynomial_identity · Uses: G001, S015 ## Reverse checks - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute b add coefficient0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute b multiply before degree0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute b add coefficient1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute b multiply before degree1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute b add coefficient2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute b multiply before degree2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute a add coefficient0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute a multiply before degree0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute a add coefficient1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute a multiply before degree1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute a add coefficient2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute a multiply before degree2 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. ## Result (-34, 41) 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: 1d07f27f45deb0bcc67bb5e97f37ec3455824e3abbee88b42057946ff253f80f - candidate record SHA-256: 03916f5acfe811fe3534414b86b2e29dce727db0677b91bd26acaa8352085bd4 - 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-0053.json)