Question 32696: Signed Moment Integral of a Polynomial

Original question and givens

On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=-3, b=1. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=-3, b=1. 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 = [[8, 21, 1, 1]]

derived = [[8, 21, 1, 1]]

Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 := mul(1, -3)

Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 = -3

source roles = [Polynomial ORIGINAL coefficient0 lower endpoint, -3, 1]

derived = [-3]

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

Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 = 1

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

derived = [1]

Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power := sub(1, -3)

Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power = 4

Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference := mul(8, 4)

Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference = 32

Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent := div(32, 1)

Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent = 32

source roles = [Polynomial ORIGINAL coefficient0, 8, 0, -3, 1]

derived = [-3, 1, 4, 32/1]

Polynomial moment ORIGINAL sum through coefficient0 := sum(0, 32)

Polynomial moment ORIGINAL sum through coefficient0 = 32

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 := mul(1, -3)

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 = -3

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 := mul(-3, -3)

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 = 9

source roles = [Polynomial ORIGINAL coefficient1 lower endpoint, -3, 2]

derived = [9]

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

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 = 1

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 := mul(1, 1)

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 = 1

source roles = [Polynomial ORIGINAL coefficient1 upper endpoint, 1, 2]

derived = [1]

Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power := sub(1, 9)

Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power = -8

Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference := mul(21, -8)

Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference = -168

Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent := div(-168, 2)

Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent = -84

source roles = [Polynomial ORIGINAL coefficient1, 21, 1, -3, 1]

derived = [9, 1, -8, -84/1]

Polynomial moment ORIGINAL sum through coefficient1 := sum(32, -84)

Polynomial moment ORIGINAL sum through coefficient1 = -52

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 := mul(1, -3)

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 = -3

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 := mul(-3, -3)

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 = 9

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 := mul(9, -3)

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 = -27

source roles = [Polynomial ORIGINAL coefficient2 lower endpoint, -3, 3]

derived = [-27]

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 := mul(1, 1)

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 = 1

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 := mul(1, 1)

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 = 1

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 := mul(1, 1)

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 = 1

source roles = [Polynomial ORIGINAL coefficient2 upper endpoint, 1, 3]

derived = [1]

Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power := sub(1, -27)

Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power = 28

Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference := mul(1, 28)

Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference = 28

Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent := div(28, 3)

Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent = 28/3

source roles = [Polynomial ORIGINAL coefficient2, 1, 2, -3, 1]

derived = [-27, 1, 28, 28/3]

Polynomial moment ORIGINAL sum through coefficient2 := sum(-52, 28/3)

Polynomial moment ORIGINAL sum through coefficient2 = -128/3

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 := mul(1, -3)

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 = -3

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 := mul(-3, -3)

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 = 9

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 := mul(9, -3)

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 = -27

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 := mul(-27, -3)

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 = 81

source roles = [Polynomial ORIGINAL coefficient3 lower endpoint, -3, 4]

derived = [81]

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 := mul(1, 1)

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 = 1

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 := mul(1, 1)

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 = 1

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 := mul(1, 1)

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 = 1

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 := mul(1, 1)

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 = 1

source roles = [Polynomial ORIGINAL coefficient3 upper endpoint, 1, 4]

derived = [1]

Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power := sub(1, 81)

Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power = -80

Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference := mul(1, -80)

Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference = -80

Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent := div(-80, 4)

Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent = -20

source roles = [Polynomial ORIGINAL coefficient3, 1, 3, -3, 1]

derived = [81, 1, -80, -20/1]

Polynomial moment ORIGINAL sum through coefficient3 := sum(-128/3, -20)

Polynomial moment ORIGINAL sum through coefficient3 = -188/3

source roles = [[8, 21, 1, 1], -3, 1]

derived = [-188/3]

Reverse checks

• S051 · 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.
• S050 · premise_and_result_audit — Recompute Polynomial moment ORIGINAL sum through coefficient3 with exact rational arithmetic, not rounded decimals.
• S049 · 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.
• S048 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals.
• S045 · 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.
• S044 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S040 · 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.
• S039 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Polynomial moment ORIGINAL sum through coefficient2 with exact rational arithmetic, not rounded decimals.
• S034 · 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.
• S033 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals.
• S030 · 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.
• S029 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• S026 · 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.
• S025 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals.
• 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

-188/3

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.