In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5]. Compute p(-11) exactly.
G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5]. Compute p(-11) 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.
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.
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.
Horner ORIGINAL prefix times x at degree0 := mul(0, -11)
After
Horner ORIGINAL prefix times x at degree0 = 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.
Horner ORIGINAL add coefficient at degree0 := sum(0, 5)
After
Horner ORIGINAL add coefficient at degree0 = 5
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.
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.
S005 · 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.
S004 · premise_and_result_audit — Recompute Horner ORIGINAL add coefficient at degree0 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Horner ORIGINAL prefix times x at degree0 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
5
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.