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Question 34282: Exact Polynomial Evaluation

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0067-34282 · q-20261003-r02-polynomial-0007

Original question and givens

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-6, 3, -1]. Compute p(-6) exactly.

  • G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-6, 3, -1]. Compute p(-6) 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001

S002 · implies

Before

source roles = [polynomial ORIGINAL, [-6, 3, -1]]

After

derived = [[-6, 3, -1], [-6, 3, -1]]

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 dependencies

Rule: Polynomial_RAW_ALL_SAFE_coefficients_trim_ONLY_trailing_zero · Uses: G001, S001

S003 · calculation

Before

Horner ORIGINAL prefix times x at degree2 := mul(0, -6)

After

Horner ORIGINAL prefix times x at degree2 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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

After

Horner ORIGINAL add coefficient at degree2 = -1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Horner ORIGINAL prefix times x at degree1 := mul(-1, -6)

After

Horner ORIGINAL prefix times x at degree1 = 6

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Horner ORIGINAL add coefficient at degree1 := sum(6, 3)

After

Horner ORIGINAL add coefficient at degree1 = 9

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Horner ORIGINAL prefix times x at degree0 := mul(9, -6)

After

Horner ORIGINAL prefix times x at degree0 = -54

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Horner ORIGINAL add coefficient at degree0 := sum(-54, -6)

After

Horner ORIGINAL add coefficient at degree0 = -60

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [[-6, 3, -1], -6]

After

derived = [-60]

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 dependencies

Rule: Polynomial_HORNER_complete_exact_evaluation_NOT_identity · Uses: G001, S008

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

-60

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: 443e58ea0a981385e85768f3f0004985a165c66885f100d14d73a203f46cd741
  • candidate record SHA-256: cf58387256d64de738413e875eb4c2aa20e92be6c8a757e1777bfbdb053ce18f
  • semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
  • question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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