Question 32693: Signed Moment Integral of a Polynomial

Original question and givens

On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=2, b=-3. Give a reduced fraction with positive denominator, retaining /1 for integers.

• G001 — On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=2, b=-3. 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 = [[5, 18, 1, 2]]

derived = [[5, 18, 1, 2]]

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, -3)

Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 = -3

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

derived = [-3]

Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power := sub(-3, 2)

Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power = -5

Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference := mul(5, -5)

Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference = -25

Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent := div(-25, 1)

Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent = -25

source roles = [Polynomial ORIGINAL coefficient0, 5, 0, 2, -3]

derived = [2, -3, -5, -25/1]

Polynomial moment ORIGINAL sum through coefficient0 := sum(0, -25)

Polynomial moment ORIGINAL sum through coefficient0 = -25

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 := mul(1, 2)

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 = 2

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 := mul(2, 2)

Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 = 4

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

derived = [4]

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

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 = -3

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

Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 = 9

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

derived = [9]

Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power := sub(9, 4)

Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power = 5

Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference := mul(18, 5)

Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference = 90

Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent := div(90, 2)

Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent = 45

source roles = [Polynomial ORIGINAL coefficient1, 18, 1, 2, -3]

derived = [4, 9, 5, 45/1]

Polynomial moment ORIGINAL sum through coefficient1 := sum(-25, 45)

Polynomial moment ORIGINAL sum through coefficient1 = 20

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 := mul(1, 2)

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 = 2

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 := mul(2, 2)

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 = 4

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 := mul(4, 2)

Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 = 8

source roles = [Polynomial ORIGINAL coefficient2 lower endpoint, 2, 3]

derived = [8]

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

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 = -3

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

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 = 9

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

Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 = -27

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

derived = [-27]

Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power := sub(-27, 8)

Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power = -35

Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference := mul(1, -35)

Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference = -35

Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent := div(-35, 3)

Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent = -35/3

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

derived = [8, -27, -35, -35/3]

Polynomial moment ORIGINAL sum through coefficient2 := sum(20, -35/3)

Polynomial moment ORIGINAL sum through coefficient2 = 25/3

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 := mul(1, 2)

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 = 2

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 := mul(2, 2)

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 = 4

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 := mul(4, 2)

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 = 8

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 := mul(8, 2)

Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 = 16

source roles = [Polynomial ORIGINAL coefficient3 lower endpoint, 2, 4]

derived = [16]

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

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 = -3

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

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 = 9

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

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 = -27

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

Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 = 81

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

derived = [81]

Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power := sub(81, 16)

Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power = 65

Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference := mul(2, 65)

Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference = 130

Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent := div(130, 4)

Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent = 65/2

source roles = [Polynomial ORIGINAL coefficient3, 2, 3, 2, -3]

derived = [16, 81, 65, 65/2]

Polynomial moment ORIGINAL sum through coefficient3 := sum(25/3, 65/2)

Polynomial moment ORIGINAL sum through coefficient3 = 245/6

source roles = [[5, 18, 1, 2], 2, -3]

derived = [245/6]

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

245/6

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.