Question 24060: A Declared Shifted Polynomial Reconstructs Exactly from Its Finite Jet

Original question and givens

Use shifted_coefficients=[10, 20, 30, 40] of the explicit polynomial sum c_k*(x−a)^k, center a=-3, evaluation point=-1. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

• G001 — Use shifted_coefficients=[10, 20, 30, 40] of the explicit polynomial sum c_k*(x−a)^k, center a=-3, evaluation point=-1. Give (shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate).

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-taylor-polynomial

Step-by-step derivation

source roles = [taylor-exact-polynomial-reconstruction]

derived = [[shifted_coefficients, derivatives_at_center, reconstructed_coefficients, point, original_value, reconstructed_value, equal, certificate]]

source roles = [-3, [10, 20, 30, 40]]

derived = [P(x)=sum c_k*(x-center)^k, 3]

source roles = [0]

derived = [1]

Taylor derivative0 from shifted coefficient := mul(10, 1)

Taylor derivative0 from shifted coefficient = 10

Taylor recover coefficient0 := div(10, 1)

Taylor recover coefficient0 = 10

Taylor k1 factorial step1 := mul(1, 1)

Taylor k1 factorial step1 = 1

Taylor derivative1 from shifted coefficient := mul(20, 1)

Taylor derivative1 from shifted coefficient = 20

Taylor recover coefficient1 := div(20, 1)

Taylor recover coefficient1 = 20

Taylor k2 factorial step1 := mul(1, 1)

Taylor k2 factorial step1 = 1

Taylor k2 factorial step2 := mul(1, 2)

Taylor k2 factorial step2 = 2

Taylor derivative2 from shifted coefficient := mul(30, 2)

Taylor derivative2 from shifted coefficient = 60

Taylor recover coefficient2 := div(60, 2)

Taylor recover coefficient2 = 30

Taylor k3 factorial step1 := mul(1, 1)

Taylor k3 factorial step1 = 1

Taylor k3 factorial step2 := mul(1, 2)

Taylor k3 factorial step2 = 2

Taylor k3 factorial step3 := mul(2, 3)

Taylor k3 factorial step3 = 6

Taylor derivative3 from shifted coefficient := mul(40, 6)

Taylor derivative3 from shifted coefficient = 240

Taylor recover coefficient3 := div(240, 6)

Taylor recover coefficient3 = 40

Taylor SAME point minus declared center := sub(-1, -3)

Taylor SAME point minus declared center = 2

Taylor original Horner product3 := mul(0, 2)

Taylor original Horner product3 = 0

Taylor original Horner sum3 := sum(0, 40)

Taylor original Horner sum3 = 40

Taylor original Horner product2 := mul(40, 2)

Taylor original Horner product2 = 80

Taylor original Horner sum2 := sum(80, 30)

Taylor original Horner sum2 = 110

Taylor original Horner product1 := mul(110, 2)

Taylor original Horner product1 = 220

Taylor original Horner sum1 := sum(220, 20)

Taylor original Horner sum1 = 240

Taylor original Horner product0 := mul(240, 2)

Taylor original Horner product0 = 480

Taylor original Horner sum0 := sum(480, 10)

Taylor original Horner sum0 = 490

Taylor reconstructed Horner product3 := mul(0, 2)

Taylor reconstructed Horner product3 = 0

Taylor reconstructed Horner sum3 := sum(0, 40)

Taylor reconstructed Horner sum3 = 40

Taylor reconstructed Horner product2 := mul(40, 2)

Taylor reconstructed Horner product2 = 80

Taylor reconstructed Horner sum2 := sum(80, 30)

Taylor reconstructed Horner sum2 = 110

Taylor reconstructed Horner product1 := mul(110, 2)

Taylor reconstructed Horner product1 = 220

Taylor reconstructed Horner sum1 := sum(220, 20)

Taylor reconstructed Horner sum1 = 240

Taylor reconstructed Horner product0 := mul(240, 2)

Taylor reconstructed Horner product0 = 480

Taylor reconstructed Horner sum0 := sum(480, 10)

Taylor reconstructed Horner sum0 = 490

source roles = [[10, 20, 30, 40], [10, 20, 30, 40], -3, -1]

derived = [Yes, 490, 490, Yes]

Reverse checks

• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum0 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Taylor reconstructed Horner product0 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum1 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Taylor reconstructed Horner product1 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum2 with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Taylor reconstructed Horner product2 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Taylor reconstructed Horner sum3 with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Taylor reconstructed Horner product3 with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Taylor original Horner sum0 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Taylor original Horner product0 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Taylor original Horner sum1 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Taylor original Horner product1 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Taylor original Horner sum2 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Taylor original Horner product2 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Taylor original Horner sum3 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Taylor original Horner product3 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Taylor SAME point minus declared center with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Taylor recover coefficient3 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Taylor derivative3 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Taylor k3 factorial step3 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Taylor k3 factorial step2 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Taylor k3 factorial step1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Taylor recover coefficient2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Taylor derivative2 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Taylor k2 factorial step2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Taylor k2 factorial step1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Taylor recover coefficient1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Taylor derivative1 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Taylor k1 factorial step1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Taylor recover coefficient0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Taylor derivative0 from shifted coefficient with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([10, 20, 30, 40], [10, 20, 60, 240], [10, 20, 30, 40], -1, 490, 490, Yes, degree_at_most_n_polynomial_is_exactly_reconstructed)

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.