Question 28091: An Exact Root Stops before Any Newton Division

Original question and givens

Coefficients in low-to-high degree order are=[-16, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=2. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

• G001 — Coefficients in low-to-high degree order are=[-16, 0, 1]; all iterations use exact rational arithmetic. The current iterate is x=2. Give (coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention).

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-newton-method

Step-by-step derivation

source roles = [qf-mod50-newton-step, newton-step]

derived = [[coefficients_low_to_high, current_iterate, function_value, derivative_value, status, next_iterate_or_null, root_first_convention]]

Newton ORIGINAL polynomial ORIGINAL derivative coefficient1 := mul(1, 0)

Newton ORIGINAL polynomial ORIGINAL derivative coefficient1 = 0

Newton ORIGINAL polynomial ORIGINAL derivative coefficient2 := mul(2, 1)

Newton ORIGINAL polynomial ORIGINAL derivative coefficient2 = 2

source roles = [Newton ORIGINAL polynomial, [-16, 0, 1]]

derived = [[-16, 0, 1], [0, 2]]

Newton ORIGINAL step f ORIGINAL Horner multiply degree2 := mul(0, 2)

Newton ORIGINAL step f ORIGINAL Horner multiply degree2 = 0

Newton ORIGINAL step f ORIGINAL Horner add coefficient2 := sum(0, 1)

Newton ORIGINAL step f ORIGINAL Horner add coefficient2 = 1

Newton ORIGINAL step f ORIGINAL Horner multiply degree1 := mul(1, 2)

Newton ORIGINAL step f ORIGINAL Horner multiply degree1 = 2

Newton ORIGINAL step f ORIGINAL Horner add coefficient1 := sum(2, 0)

Newton ORIGINAL step f ORIGINAL Horner add coefficient1 = 2

Newton ORIGINAL step f ORIGINAL Horner multiply degree0 := mul(2, 2)

Newton ORIGINAL step f ORIGINAL Horner multiply degree0 = 4

Newton ORIGINAL step f ORIGINAL Horner add coefficient0 := sum(4, -16)

Newton ORIGINAL step f ORIGINAL Horner add coefficient0 = -12

Newton ORIGINAL step fprime ORIGINAL Horner multiply degree1 := mul(0, 2)

Newton ORIGINAL step fprime ORIGINAL Horner multiply degree1 = 0

Newton ORIGINAL step fprime ORIGINAL Horner add coefficient1 := sum(0, 2)

Newton ORIGINAL step fprime ORIGINAL Horner add coefficient1 = 2

Newton ORIGINAL step fprime ORIGINAL Horner multiply degree0 := mul(2, 2)

Newton ORIGINAL step fprime ORIGINAL Horner multiply degree0 = 4

Newton ORIGINAL step fprime ORIGINAL Horner add coefficient0 := sum(4, 0)

Newton ORIGINAL step fprime ORIGINAL Horner add coefficient0 = 4

Newton ORIGINAL step ORIGINAL f over fprime := div(-12, 4)

Newton ORIGINAL step ORIGINAL f over fprime = -3

Newton ORIGINAL step ORIGINAL tangent zero update := sub(2, -3)

Newton ORIGINAL step ORIGINAL tangent zero update = 5

source roles = [Newton ORIGINAL step, [-16, 0, 1], [0, 2], 2]

derived = [-12, 4, updated, 5]

Reverse checks

• S017 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S016 · premise_and_result_audit — Recompute Newton ORIGINAL step ORIGINAL tangent zero update with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Newton ORIGINAL step ORIGINAL f over fprime with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Newton ORIGINAL step fprime ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Newton ORIGINAL step fprime ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Newton ORIGINAL step fprime ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Newton ORIGINAL step fprime ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Newton ORIGINAL step f ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Newton ORIGINAL step f ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Newton ORIGINAL step f ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Newton ORIGINAL step f ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Newton ORIGINAL step f ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Newton ORIGINAL step f ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S003 · premise_and_result_audit — Recompute Newton ORIGINAL polynomial ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Newton ORIGINAL polynomial ORIGINAL derivative coefficient1 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.

Result

([-16, 0, 1], 2, -12, 4, updated, 5, root_test_precedes_division)

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.