Question 34293: Exact Polynomial Evaluation

Original question and givens

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5, 0, 1]. Compute p(5) exactly.

• G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5, 0, 1]. Compute p(5) 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

Step-by-step derivation

source roles = [qf-polynomial-exact-evaluation, [coefficients, operation, x]]

derived = [evaluate, no-invented-domain]

source roles = [polynomial ORIGINAL, [5, 0, 1]]

derived = [[5, 0, 1], [5, 0, 1]]

Horner ORIGINAL prefix times x at degree2 := mul(0, 5)

Horner ORIGINAL prefix times x at degree2 = 0

Horner ORIGINAL add coefficient at degree2 := sum(0, 1)

Horner ORIGINAL add coefficient at degree2 = 1

Horner ORIGINAL prefix times x at degree1 := mul(1, 5)

Horner ORIGINAL prefix times x at degree1 = 5

Horner ORIGINAL add coefficient at degree1 := sum(5, 0)

Horner ORIGINAL add coefficient at degree1 = 5

Horner ORIGINAL prefix times x at degree0 := mul(5, 5)

Horner ORIGINAL prefix times x at degree0 = 25

Horner ORIGINAL add coefficient at degree0 := sum(25, 5)

Horner ORIGINAL add coefficient at degree0 = 30

source roles = [[5, 0, 1], 5]

derived = [30]

Reverse checks

• S009 · 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.
• S008 · premise_and_result_audit — Recompute Horner ORIGINAL add coefficient at degree0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Horner ORIGINAL prefix times x at degree0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Horner ORIGINAL add coefficient at degree1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Horner ORIGINAL prefix times x at degree1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Horner ORIGINAL add coefficient at degree2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Horner ORIGINAL prefix times x at degree2 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

30

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.