Question 1330: Zero-Constant Antiderivative of a Polynomial

Original question and givens

On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

• G001 — On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

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-antiderivative

Step-by-step derivation

domain=R; coefficient lists are in ascending powers

constant term first; rational pairs=[numerator,denominator]

f coefficients ascending=[-8, 5, 0, 1]

seek F₀ with constant coefficient0 and F₀′=f

primitive coefficient at degree1 := div(-8, 1)

primitive coefficient at degree1 = -8

differentiate back coefficient0 := mul(-8, 1)

differentiate back coefficient0 = -8

primitive coefficient at degree2 := div(5, 2)

primitive coefficient at degree2 = 5/2

differentiate back coefficient1 := mul(5/2, 2)

differentiate back coefficient1 = 5

primitive coefficient at degree3 := div(0, 3)

primitive coefficient at degree3 = 0

differentiate back coefficient2 := mul(0, 3)

differentiate back coefficient2 = 0

primitive coefficient at degree4 := div(1, 4)

primitive coefficient at degree4 = 1/4

differentiate back coefficient3 := mul(1/4, 4)

differentiate back coefficient3 = 1

[0, -8, 5/2, 0, 1/4]

all derivative coefficients agree with f

[0, -8, 5/2, 0, 1/4]

reduced coefficients;integer coefficients retain /1

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute differentiate back coefficient3 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute primitive coefficient at degree4 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute differentiate back coefficient2 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute primitive coefficient at degree3 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute differentiate back coefficient1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute primitive coefficient at degree2 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute differentiate back coefficient0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute primitive coefficient at degree1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

[0/1, -8/1, 5/2, 0/1, 1/4]

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.