# Question 34289: Exact Polynomial Evaluation [Return to catalogue page 172](https://ocme.evemisslab.com/en/questions/page/172/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-r02-polynomial-foundations/#q-20261003-r02-polynomial-0014) 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. R0067-34289 · q-20261003-r02-polynomial-0014 ## Original question and givens In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [1, 3]. Compute p(1) exactly. - G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [1, 3]. Compute p(1) exactly. ## 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 source roles = [qf-polynomial-exact-evaluation, [coefficients, operation, x]] After derived = [evaluate, no-invented-domain] Conditions: none Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations. Step identifiers and dependenciesRule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001 S002 · impliesBefore source roles = [polynomial ORIGINAL, [1, 3]] After derived = [[1, 3], [1, 3]] Conditions: none In Q[x],coefficients run from low to high degree.Type EVERY RAW SAFE cell of a0..32-entry list,including trailing zeros.Trim ONLY trailing zeros,not internal zeros.The zero polynomial has empty internal support but displays[0];keep the ORIGINAL source vector unchanged. Step identifiers and dependenciesRule: Polynomial_RAW_ALL_SAFE_coefficients_trim_ONLY_trailing_zero · Uses: G001, S001 S003 · calculationBefore Horner ORIGINAL prefix times x at degree1 := mul(0, 1) After Horner ORIGINAL prefix times x at degree1 = 0 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Horner ORIGINAL add coefficient at degree1 := sum(0, 3) After Horner ORIGINAL add coefficient at degree1 = 3 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Horner ORIGINAL prefix times x at degree0 := mul(3, 1) After Horner ORIGINAL prefix times x at degree0 = 3 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Horner ORIGINAL add coefficient at degree0 := sum(3, 1) After Horner ORIGINAL add coefficient at degree0 = 4 Conditions: none Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [[1, 3], 1] After derived = [4] Conditions: none Starting at the highest power,repeatedly multiply by x and add THIS coefficient,covering ALL coefficients.The zero polynomial’s empty loop gives0.This evaluates THIS input only;one-point agreement is not a polynomial identity.Derived integers remain unbounded. Step identifiers and dependenciesRule: Polynomial_HORNER_complete_exact_evaluation_NOT_identity · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S006 · premise_and_result_audit — Recompute Horner ORIGINAL add coefficient at degree0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Horner ORIGINAL prefix times x at degree0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Horner ORIGINAL add coefficient at degree1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Horner ORIGINAL prefix times x at degree1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. - S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof. ## Result 4 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: 31b92e465aa9ee4f364578b6795585999f1a89efa99a7ce42c890837f22b00fb - candidate record SHA-256: dfe88dfb2abb60dfcfa7dfb13c6cb426f70b3ee693ca9dddb33e10566aaa7296 - semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c - question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r02-polynomial-0014.json)