Question 32677: Signed Moment Integral of a Polynomial

Original question and givens

On all R, f has ascending coefficients [3], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=2, b=-2. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f has ascending coefficients [3], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=2, b=-2. 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 = [[3]]

derived = [[3]]

Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 := mul(1, 2)

Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 = 2

source roles = [Polynomial ORIGINAL coefficient0 lower endpoint, 2, 1]

derived = [2]

Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 := mul(1, -2)

Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 = -2

source roles = [Polynomial ORIGINAL coefficient0 upper endpoint, -2, 1]

derived = [-2]

Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power := sub(-2, 2)

Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power = -4

Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference := mul(3, -4)

Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference = -12

Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent := div(-12, 1)

Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent = -12

source roles = [Polynomial ORIGINAL coefficient0, 3, 0, 2, -2]

derived = [2, -2, -4, -12/1]

Polynomial moment ORIGINAL sum through coefficient0 := sum(0, -12)

Polynomial moment ORIGINAL sum through coefficient0 = -12

source roles = [[3], 2, -2]

derived = [-12/1]

Reverse checks

• S012 · 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.
• 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

-12/1

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.