# Question 26487: A Polynomial of Degree at Most the Taylor Order Has Zero Remainder [Return to catalogue page 133](https://ocme.evemisslab.com/en/questions/page/133/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261116-r03-taylor-remainder-theorem-foundations/#q-20261015-advance-20261116-r03-taylor-remainder-theorem-0037) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0051-26487 · q-20261015-advance-20261116-r03-taylor-remainder-theorem-0037 ## Original question and givens Coefficients=[13, 8, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate). - G001 — Coefficients=[13, 8, 3] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=2; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, 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-remainder-theorem](https://ocme.evemisslab.com/en/mko/mko-taylor-remainder-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [taylor-zero-remainder] After derived = [[degree, order, Taylor_terms, remainder, zero_remainder, certificate]] Conditions: none List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Taylor ORIGINAL x minus declared center := sub(1/2, 0) After Taylor ORIGINAL x minus declared center = 1/2 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore (0)^0 After Taylor ORIGINAL derivative0 center power0 term0 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S002 S004 · calculationBefore Taylor ORIGINAL derivative0 falling factor times power term0 := mul(1, 1) After Taylor ORIGINAL derivative0 falling factor times power term0 = 1 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 := mul(13, 1) After Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 13 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 13) After Taylor ORIGINAL derivative0 ascending derivative sum term0 = 13 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore (0)^1 After Taylor ORIGINAL derivative0 center power1 term1 = 0 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S006 S008 · calculationBefore Taylor ORIGINAL derivative0 falling factor times power term1 := mul(1, 0) After Taylor ORIGINAL derivative0 falling factor times power term1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 := mul(8, 0) After Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(13, 0) After Taylor ORIGINAL derivative0 ascending derivative sum term1 = 13 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore (0)^2 After Taylor ORIGINAL derivative0 center power2 term2 = 0 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S010 S012 · calculationBefore Taylor ORIGINAL derivative0 falling factor times power term2 := mul(1, 0) After Taylor ORIGINAL derivative0 falling factor times power term2 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 := mul(3, 0) After Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(13, 0) After Taylor ORIGINAL derivative0 ascending derivative sum term2 = 13 Conditions: none Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore source roles = [0, 0, [13, 8, 3]] After derived = [[[0, 13, 1, 1, 13], [1, 8, 1, 0, 0], [2, 3, 1, 0, 0]], 13] Conditions: none The kth derivative of x^j is j!/(j−k)!*x^(j−k);terms j