Question 32676: Signed Moment Integral of a Polynomial

Original question and givens

On all R, f has ascending coefficients [1, 2], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=0, b=0. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f has ascending coefficients [1, 2], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=0, b=0. Give a reduced fraction with positive denominator, retaining /1 for 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-definite-integral

Step-by-step derivation

source roles = [qf-integral-polynomial, polynomial_integral, reals, []]

derived = [[signed_moment_fraction_RETAIN1]]

source roles = [[1, 2]]

derived = [[1, 2]]

Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 := mul(1, 0)

Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 = 0

source roles = [Polynomial ORIGINAL coefficient0 lower endpoint, 0, 1]

derived = [0]

Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 := mul(1, 0)

Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 = 0

source roles = [Polynomial ORIGINAL coefficient0 upper endpoint, 0, 1]

derived = [0]

Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power := sub(0, 0)

Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power = 0

Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference := mul(1, 0)

Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference = 0

Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent := div(0, 1)

Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent = 0

source roles = [Polynomial ORIGINAL coefficient0, 1, 0, 0, 0]

derived = [0, 0, 0, 0/1]

Polynomial moment ORIGINAL sum through coefficient0 := sum(0, 0)

Polynomial moment ORIGINAL sum through coefficient0 = 0

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 := mul(1, 0)

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 = 0

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 := mul(0, 0)

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 = 0

source roles = [Polynomial ORIGINAL coefficient1 lower endpoint, 0, 2]

derived = [0]

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 := mul(1, 0)

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 = 0

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 := mul(0, 0)

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 = 0

source roles = [Polynomial ORIGINAL coefficient1 upper endpoint, 0, 2]

derived = [0]

Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power := sub(0, 0)

Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power = 0

Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference := mul(2, 0)

Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference = 0

Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent := div(0, 2)

Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent = 0

source roles = [Polynomial ORIGINAL coefficient1, 2, 1, 0, 0]

derived = [0, 0, 0, 0/1]

Polynomial moment ORIGINAL sum through coefficient1 := sum(0, 0)

Polynomial moment ORIGINAL sum through coefficient1 = 0

source roles = [[1, 2], 0, 0]

derived = [0/1]

Reverse checks

• S023 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S022 · premise_and_result_audit — Recompute Polynomial moment ORIGINAL sum through coefficient1 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S020 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S016 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S013 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Polynomial moment ORIGINAL sum through coefficient0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S009 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S005 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S003 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
• S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

0/1

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.