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R0001-01318 · q-20261007-advance-20261008-r06-antiderivative-0018
On all R, f(x)=(-18)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.
G001 — On all R, f(x)=(-18)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.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.
Before
domain=R; coefficient lists are in ascending powers
After
constant term first; rational pairs=[numerator,denominator]
Conditions: none
Fix coefficient order;do not read ascending coefficients as descending or swap numerator and denominator.
Rule: polynomial_representation · Uses: G001
Before
f coefficients ascending=[0, 0, 0, 0, 0, -18]
After
seek F₀ with constant coefficient0 and F₀′=f
Conditions: none
Antiderivatives are not generally unique; this task additionally fixes the constant coefficient at0.
Rule: canonical_primitive_domain · Uses: G001, S001
Before
primitive coefficient at degree1 := div(0, 1)
After
primitive coefficient at degree1 = 0
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
differentiate back coefficient0 := mul(0, 1)
After
differentiate back coefficient0 = 0
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
primitive coefficient at degree2 := div(0, 2)
After
primitive coefficient at degree2 = 0
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
differentiate back coefficient1 := mul(0, 2)
After
differentiate back coefficient1 = 0
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
primitive coefficient at degree3 := div(0, 3)
After
primitive coefficient at degree3 = 0
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
differentiate back coefficient2 := mul(0, 3)
After
differentiate back coefficient2 = 0
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
primitive coefficient at degree4 := div(0, 4)
After
primitive coefficient at degree4 = 0
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
differentiate back coefficient3 := mul(0, 4)
After
differentiate back coefficient3 = 0
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
primitive coefficient at degree5 := div(0, 5)
After
primitive coefficient at degree5 = 0
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
differentiate back coefficient4 := mul(0, 5)
After
differentiate back coefficient4 = 0
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
primitive coefficient at degree6 := div(-18, 6)
After
primitive coefficient at degree6 = -3
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
differentiate back coefficient5 := mul(-3, 6)
After
differentiate back coefficient5 = -18
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[0, 0, 0, 0, 0, 0, -3]
After
all derivative coefficients agree with f
Conditions: none
Coefficient agreement proves the polynomial identity on all reals,not merely equality at a few points.
Rule: global_polynomial_identity · Uses: G001, S014
Before
[0, 0, 0, 0, 0, 0, -3]
After
reduced coefficients;integer coefficients retain /1
Conditions: none
Return ascending coefficients,trim redundant leading-in-degree zeros,and retain[0/1] for the zero polynomial.
Rule: coefficient_output_protocol · Uses: G001, S015
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Recompute differentiate back coefficient5 with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute primitive coefficient at degree6 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute differentiate back coefficient4 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute primitive coefficient at degree5 with exact rational arithmetic, not rounded decimals.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.[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -3/1]
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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