Question 26490: A Polynomial of Degree at Most the Taylor Order Has Zero Remainder

Original question and givens

Coefficients=[16, 14, 12, 10, 8, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, Taylor_terms, remainder, zero_remainder, certificate).

• G001 — Coefficients=[16, 14, 12, 10, 8, 6] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=5; 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

Step-by-step derivation

source roles = [taylor-zero-remainder]

derived = [[degree, order, Taylor_terms, remainder, zero_remainder, certificate]]

Taylor ORIGINAL x minus declared center := sub(1/2, 0)

Taylor ORIGINAL x minus declared center = 1/2

(0)^0

Taylor ORIGINAL derivative0 center power0 term0 = 1

Taylor ORIGINAL derivative0 falling factor times power term0 := mul(1, 1)

Taylor ORIGINAL derivative0 falling factor times power term0 = 1

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 := mul(16, 1)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 16

Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 16)

Taylor ORIGINAL derivative0 ascending derivative sum term0 = 16

(0)^1

Taylor ORIGINAL derivative0 center power1 term1 = 0

Taylor ORIGINAL derivative0 falling factor times power term1 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term1 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 := mul(14, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(16, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term1 = 16

(0)^2

Taylor ORIGINAL derivative0 center power2 term2 = 0

Taylor ORIGINAL derivative0 falling factor times power term2 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term2 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 := mul(12, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(16, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term2 = 16

(0)^3

Taylor ORIGINAL derivative0 center power3 term3 = 0

Taylor ORIGINAL derivative0 falling factor times power term3 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term3 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 := mul(10, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term3 := sum(16, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term3 = 16

(0)^4

Taylor ORIGINAL derivative0 center power4 term4 = 0

Taylor ORIGINAL derivative0 falling factor times power term4 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term4 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 := mul(8, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term4 := sum(16, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term4 = 16

(0)^5

Taylor ORIGINAL derivative0 center power5 term5 = 0

Taylor ORIGINAL derivative0 falling factor times power term5 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term5 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 := mul(6, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term5 := sum(16, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term5 = 16

source roles = [0, 0, [16, 14, 12, 10, 8, 6]]

derived = [[[0, 16, 1, 1, 16], [1, 14, 1, 0, 0], [2, 12, 1, 0, 0], [3, 10, 1, 0, 0], [4, 8, 1, 0, 0], [5, 6, 1, 0, 0]], 16]

(1/2)^0

Taylor ORIGINAL displacement power0 = 1

Taylor ORIGINAL derivative times displacement0 := mul(16, 1)

Taylor ORIGINAL derivative times displacement0 = 16

Taylor ORIGINAL term0 divide factorial := div(16, 1)

Taylor ORIGINAL term0 divide factorial = 16

(0)^0

Taylor ORIGINAL derivative1 center power0 term1 = 1

Taylor ORIGINAL derivative1 falling factor times power term1 := mul(1, 1)

Taylor ORIGINAL derivative1 falling factor times power term1 = 1

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 := mul(14, 1)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 = 14

Taylor ORIGINAL derivative1 ascending derivative sum term1 := sum(0, 14)

Taylor ORIGINAL derivative1 ascending derivative sum term1 = 14

(0)^1

Taylor ORIGINAL derivative1 center power1 term2 = 0

Taylor ORIGINAL derivative1 falling factor times power term2 := mul(2, 0)

Taylor ORIGINAL derivative1 falling factor times power term2 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 := mul(12, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term2 := sum(14, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term2 = 14

(0)^2

Taylor ORIGINAL derivative1 center power2 term3 = 0

Taylor ORIGINAL derivative1 falling factor times power term3 := mul(3, 0)

Taylor ORIGINAL derivative1 falling factor times power term3 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 := mul(10, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term3 := sum(14, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term3 = 14

(0)^3

Taylor ORIGINAL derivative1 center power3 term4 = 0

Taylor ORIGINAL derivative1 falling factor times power term4 := mul(4, 0)

Taylor ORIGINAL derivative1 falling factor times power term4 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 := mul(8, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term4 := sum(14, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term4 = 14

(0)^4

Taylor ORIGINAL derivative1 center power4 term5 = 0

Taylor ORIGINAL derivative1 falling factor times power term5 := mul(5, 0)

Taylor ORIGINAL derivative1 falling factor times power term5 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 := mul(6, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term5 := sum(14, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term5 = 14

source roles = [1, 0, [16, 14, 12, 10, 8, 6]]

derived = [[[1, 14, 1, 1, 14], [2, 12, 2, 0, 0], [3, 10, 3, 0, 0], [4, 8, 4, 0, 0], [5, 6, 5, 0, 0]], 14]

(1/2)^1

Taylor ORIGINAL displacement power1 = 1/2

Taylor ORIGINAL derivative times displacement1 := mul(14, 1/2)

Taylor ORIGINAL derivative times displacement1 = 7

Taylor ORIGINAL term1 divide factorial := div(7, 1)

Taylor ORIGINAL term1 divide factorial = 7

(0)^0

Taylor ORIGINAL derivative2 center power0 term2 = 1

Taylor ORIGINAL derivative2 falling factor times power term2 := mul(2, 1)

Taylor ORIGINAL derivative2 falling factor times power term2 = 2

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 := mul(12, 2)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 = 24

Taylor ORIGINAL derivative2 ascending derivative sum term2 := sum(0, 24)

Taylor ORIGINAL derivative2 ascending derivative sum term2 = 24

(0)^1

Taylor ORIGINAL derivative2 center power1 term3 = 0

Taylor ORIGINAL derivative2 falling factor times power term3 := mul(6, 0)

Taylor ORIGINAL derivative2 falling factor times power term3 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 := mul(10, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term3 := sum(24, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term3 = 24

(0)^2

Taylor ORIGINAL derivative2 center power2 term4 = 0

Taylor ORIGINAL derivative2 falling factor times power term4 := mul(12, 0)

Taylor ORIGINAL derivative2 falling factor times power term4 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 := mul(8, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term4 := sum(24, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term4 = 24

(0)^3

Taylor ORIGINAL derivative2 center power3 term5 = 0

Taylor ORIGINAL derivative2 falling factor times power term5 := mul(20, 0)

Taylor ORIGINAL derivative2 falling factor times power term5 = 0

Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 := mul(6, 0)

Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative2 ascending derivative sum term5 := sum(24, 0)

Taylor ORIGINAL derivative2 ascending derivative sum term5 = 24

source roles = [2, 0, [16, 14, 12, 10, 8, 6]]

derived = [[[2, 12, 2, 1, 24], [3, 10, 6, 0, 0], [4, 8, 12, 0, 0], [5, 6, 20, 0, 0]], 24]

(1/2)^2

Taylor ORIGINAL displacement power2 = 1/4

Taylor ORIGINAL derivative times displacement2 := mul(24, 1/4)

Taylor ORIGINAL derivative times displacement2 = 6

Taylor ORIGINAL term2 divide factorial := div(6, 2)

Taylor ORIGINAL term2 divide factorial = 3

(0)^0

Taylor ORIGINAL derivative3 center power0 term3 = 1

Taylor ORIGINAL derivative3 falling factor times power term3 := mul(6, 1)

Taylor ORIGINAL derivative3 falling factor times power term3 = 6

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 := mul(10, 6)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 = 60

Taylor ORIGINAL derivative3 ascending derivative sum term3 := sum(0, 60)

Taylor ORIGINAL derivative3 ascending derivative sum term3 = 60

(0)^1

Taylor ORIGINAL derivative3 center power1 term4 = 0

Taylor ORIGINAL derivative3 falling factor times power term4 := mul(24, 0)

Taylor ORIGINAL derivative3 falling factor times power term4 = 0

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 := mul(8, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term4 := sum(60, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term4 = 60

(0)^2

Taylor ORIGINAL derivative3 center power2 term5 = 0

Taylor ORIGINAL derivative3 falling factor times power term5 := mul(60, 0)

Taylor ORIGINAL derivative3 falling factor times power term5 = 0

Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 := mul(6, 0)

Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative3 ascending derivative sum term5 := sum(60, 0)

Taylor ORIGINAL derivative3 ascending derivative sum term5 = 60

source roles = [3, 0, [16, 14, 12, 10, 8, 6]]

derived = [[[3, 10, 6, 1, 60], [4, 8, 24, 0, 0], [5, 6, 60, 0, 0]], 60]

(1/2)^3

Taylor ORIGINAL displacement power3 = 1/8

Taylor ORIGINAL derivative times displacement3 := mul(60, 1/8)

Taylor ORIGINAL derivative times displacement3 = 15/2

Taylor ORIGINAL term3 divide factorial := div(15/2, 6)

Taylor ORIGINAL term3 divide factorial = 5/4

(0)^0

Taylor ORIGINAL derivative4 center power0 term4 = 1

Taylor ORIGINAL derivative4 falling factor times power term4 := mul(24, 1)

Taylor ORIGINAL derivative4 falling factor times power term4 = 24

Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 := mul(8, 24)

Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 = 192

Taylor ORIGINAL derivative4 ascending derivative sum term4 := sum(0, 192)

Taylor ORIGINAL derivative4 ascending derivative sum term4 = 192

(0)^1

Taylor ORIGINAL derivative4 center power1 term5 = 0

Taylor ORIGINAL derivative4 falling factor times power term5 := mul(120, 0)

Taylor ORIGINAL derivative4 falling factor times power term5 = 0

Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 := mul(6, 0)

Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 = 0

Taylor ORIGINAL derivative4 ascending derivative sum term5 := sum(192, 0)

Taylor ORIGINAL derivative4 ascending derivative sum term5 = 192

source roles = [4, 0, [16, 14, 12, 10, 8, 6]]

derived = [[[4, 8, 24, 1, 192], [5, 6, 120, 0, 0]], 192]

(1/2)^4

Taylor ORIGINAL displacement power4 = 1/16

Taylor ORIGINAL derivative times displacement4 := mul(192, 1/16)

Taylor ORIGINAL derivative times displacement4 = 12

Taylor ORIGINAL term4 divide factorial := div(12, 24)

Taylor ORIGINAL term4 divide factorial = 1/2

(0)^0

Taylor ORIGINAL derivative5 center power0 term5 = 1

Taylor ORIGINAL derivative5 falling factor times power term5 := mul(120, 1)

Taylor ORIGINAL derivative5 falling factor times power term5 = 120

Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 := mul(6, 120)

Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 = 720

Taylor ORIGINAL derivative5 ascending derivative sum term5 := sum(0, 720)

Taylor ORIGINAL derivative5 ascending derivative sum term5 = 720

source roles = [5, 0, [16, 14, 12, 10, 8, 6]]

derived = [[[5, 6, 120, 1, 720]], 720]

(1/2)^5

Taylor ORIGINAL displacement power5 = 1/32

Taylor ORIGINAL derivative times displacement5 := mul(720, 1/32)

Taylor ORIGINAL derivative times displacement5 = 45/2

Taylor ORIGINAL term5 divide factorial := div(45/2, 120)

Taylor ORIGINAL term5 divide factorial = 3/16

Taylor ORIGINAL partial sum0 := sum(0, 16)

Taylor ORIGINAL partial sum0 = 16

Taylor ORIGINAL partial sum1 := sum(16, 7)

Taylor ORIGINAL partial sum1 = 23

Taylor ORIGINAL partial sum2 := sum(23, 3)

Taylor ORIGINAL partial sum2 = 26

Taylor ORIGINAL partial sum3 := sum(26, 5/4)

Taylor ORIGINAL partial sum3 = 109/4

Taylor ORIGINAL partial sum4 := sum(109/4, 1/2)

Taylor ORIGINAL partial sum4 = 111/4

Taylor ORIGINAL partial sum5 := sum(111/4, 3/16)

Taylor ORIGINAL partial sum5 = 447/16

(1/2)^0

Taylor ORIGINAL full function at x ORIGINAL argument power0 = 1

Taylor ORIGINAL full function at x coefficient term0 := mul(16, 1)

Taylor ORIGINAL full function at x coefficient term0 = 16

Taylor ORIGINAL full function at x ascending partial sum0 := sum(0, 16)

Taylor ORIGINAL full function at x ascending partial sum0 = 16

(1/2)^1

Taylor ORIGINAL full function at x ORIGINAL argument power1 = 1/2

Taylor ORIGINAL full function at x coefficient term1 := mul(14, 1/2)

Taylor ORIGINAL full function at x coefficient term1 = 7

Taylor ORIGINAL full function at x ascending partial sum1 := sum(16, 7)

Taylor ORIGINAL full function at x ascending partial sum1 = 23

(1/2)^2

Taylor ORIGINAL full function at x ORIGINAL argument power2 = 1/4

Taylor ORIGINAL full function at x coefficient term2 := mul(12, 1/4)

Taylor ORIGINAL full function at x coefficient term2 = 3

Taylor ORIGINAL full function at x ascending partial sum2 := sum(23, 3)

Taylor ORIGINAL full function at x ascending partial sum2 = 26

(1/2)^3

Taylor ORIGINAL full function at x ORIGINAL argument power3 = 1/8

Taylor ORIGINAL full function at x coefficient term3 := mul(10, 1/8)

Taylor ORIGINAL full function at x coefficient term3 = 5/4

Taylor ORIGINAL full function at x ascending partial sum3 := sum(26, 5/4)

Taylor ORIGINAL full function at x ascending partial sum3 = 109/4

(1/2)^4

Taylor ORIGINAL full function at x ORIGINAL argument power4 = 1/16

Taylor ORIGINAL full function at x coefficient term4 := mul(8, 1/16)

Taylor ORIGINAL full function at x coefficient term4 = 1/2

Taylor ORIGINAL full function at x ascending partial sum4 := sum(109/4, 1/2)

Taylor ORIGINAL full function at x ascending partial sum4 = 111/4

(1/2)^5

Taylor ORIGINAL full function at x ORIGINAL argument power5 = 1/32

Taylor ORIGINAL full function at x coefficient term5 := mul(6, 1/32)

Taylor ORIGINAL full function at x coefficient term5 = 3/16

Taylor ORIGINAL full function at x ascending partial sum5 := sum(111/4, 3/16)

Taylor ORIGINAL full function at x ascending partial sum5 = 447/16

Taylor ORIGINAL f(x) minus Taylor value := sub(447/16, 447/16)

Taylor ORIGINAL f(x) minus Taylor value = 0

source roles = [[16, 14, 12, 10, 8, 6], 0, 1/2, 5, [16, 14, 24, 60, 192, 720], [16, 7, 3, 5/4, 1/2, 3/16]]

derived = [447/16, 447/16, 0, 5]

Reverse checks

• S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S135 · premise_and_result_audit — Recompute Taylor ORIGINAL f(x) minus Taylor value with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum5 with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term5 with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S131 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum4 with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term4 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S128 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum3 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term3 with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S125 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum2 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term2 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S122 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum1 with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term1 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S119 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum0 with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term0 with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S116 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum5 with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum4 with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum3 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum2 with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum1 with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum0 with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Taylor ORIGINAL term5 divide factorial with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement5 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S107 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S106 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative5 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Recompute Taylor ORIGINAL term4 divide factorial with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement4 with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S094 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative4 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Recompute Taylor ORIGINAL term3 divide factorial with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement3 with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S078 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative3 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Recompute Taylor ORIGINAL term2 divide factorial with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement2 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative2 falling factor times power term2 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Recompute Taylor ORIGINAL term1 divide factorial with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement1 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S046 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term2 with exact rational arithmetic, not rounded decimals.
• 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 ORIGINAL derivative1 ascending derivative sum term1 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term1 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S030 · premise_and_result_audit — Recompute Taylor ORIGINAL term0 divide factorial with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement0 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term5 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term5 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term5 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term4 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term4 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term4 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term3 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term3 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term3 with exact rational arithmetic, not rounded decimals.
• 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 Taylor ORIGINAL derivative0 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term2 with exact rational arithmetic, not rounded decimals.
• 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 Taylor ORIGINAL derivative0 ascending derivative sum term1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term0 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 — Recompute Taylor ORIGINAL x minus declared center with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(5, 5, [16, 7, 3, 5/4, 1/2, 3/16], 0, Yes, degree_at_most_n_polynomial_has_zero_remainder)

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.