# Question 4270: Complete Polynomial Translation Coefficients [Return to catalogue page 22](https://ocme.evemisslab.com/en/questions/page/22/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r01-polynomial-foundations/#q-20261008-advance-20261010-r01-polynomial-0095) 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. R0007-04270 · q-20261008-advance-20261010-r01-polynomial-0095 ## Original question and givens Integer polynomials have ascending coefficients f=[7, 20, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point. - G001 — Integer polynomials have ascending coefficients f=[7, 20, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point. ## 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-polynomial](https://ocme.evemisslab.com/en/mko/mko-polynomial/) ## Step-by-step derivation S001 · impliesBefore f ascending coefficients=[7, 20, -2, 1] After coefficient k multiplies x^k;keep all required degrees Conditions: none Coefficients are in ascending powers with the constant at index0.Full coefficient operations are not identities inferred from one-point agreement;retain the original coefficient list. Step identifiers and dependenciesRule: ascending_polynomial_identity · Uses: G001 S002 · impliesBefore term 7(x+(2))^0 After expand (7)·Σ_j C(0,j)x^j(2)^(0−j) Conditions: none Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient. Step identifiers and dependenciesRule: shifted_power_binomial_identity · Uses: G001, S001 S003 · impliesBefore (2)^0 After shift power0,0 = 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 shift scaled binomial0,0 := mul(7, 1) After shift scaled binomial0,0 = 7 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore shift contribution0,0 := mul(7, 1) After shift contribution0,0 = 7 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore term 20(x+(2))^1 After expand (20)·Σ_j C(1,j)x^j(2)^(1−j) Conditions: none Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient. Step identifiers and dependenciesRule: shifted_power_binomial_identity · Uses: G001, S005 S007 · impliesBefore (2)^1 After shift power1,0 = 2 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 shift scaled binomial1,0 := mul(20, 1) After shift scaled binomial1,0 = 20 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore shift contribution1,0 := mul(20, 2) After shift contribution1,0 = 40 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore (2)^0 After shift power1,1 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S009 S011 · calculationBefore shift scaled binomial1,1 := mul(20, 1) After shift scaled binomial1,1 = 20 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore shift contribution1,1 := mul(20, 1) After shift contribution1,1 = 20 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore term -2(x+(2))^2 After expand (-2)·Σ_j C(2,j)x^j(2)^(2−j) Conditions: none Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient. Step identifiers and dependenciesRule: shifted_power_binomial_identity · Uses: G001, S012 S014 · impliesBefore (2)^2 After shift power2,0 = 4 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S013 S015 · calculationBefore shift scaled binomial2,0 := mul(-2, 1) After shift scaled binomial2,0 = -2 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore shift contribution2,0 := mul(-2, 4) After shift contribution2,0 = -8 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore (2)^1 After shift power2,1 = 2 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S016 S018 · calculationBefore shift scaled binomial2,1 := mul(-2, 2) After shift scaled binomial2,1 = -4 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore shift contribution2,1 := mul(-4, 2) After shift contribution2,1 = -8 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore (2)^0 After shift power2,2 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S019 S021 · calculationBefore shift scaled binomial2,2 := mul(-2, 1) After shift scaled binomial2,2 = -2 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore shift contribution2,2 := mul(-2, 1) After shift contribution2,2 = -2 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore term 1(x+(2))^3 After expand (1)·Σ_j C(3,j)x^j(2)^(3−j) Conditions: none Retain the original coefficient during binomial expansion and collect each resulting power j;translation does not merely change the constant coefficient. Step identifiers and dependenciesRule: shifted_power_binomial_identity · Uses: G001, S022 S024 · impliesBefore (2)^3 After shift power3,0 = 8 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S023 S025 · calculationBefore shift scaled binomial3,0 := mul(1, 1) After shift scaled binomial3,0 = 1 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore shift contribution3,0 := mul(1, 8) After shift contribution3,0 = 8 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore (2)^2 After shift power3,1 = 4 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S026 S028 · calculationBefore shift scaled binomial3,1 := mul(1, 3) After shift scaled binomial3,1 = 3 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore shift contribution3,1 := mul(3, 4) After shift contribution3,1 = 12 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore (2)^1 After shift power3,2 = 2 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S029 S031 · calculationBefore shift scaled binomial3,2 := mul(1, 3) After shift scaled binomial3,2 = 3 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore shift contribution3,2 := mul(3, 2) After shift contribution3,2 = 6 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore (2)^0 After shift power3,3 = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S032 S034 · calculationBefore shift scaled binomial3,3 := mul(1, 1) After shift scaled binomial3,3 = 1 Conditions: none Multiply this term's coefficient by its binomial coefficient. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore shift contribution3,3 := mul(1, 1) After shift contribution3,3 = 1 Conditions: none Multiply by c^(k−j) to obtain the contribution to x^j. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore shift coefficient0 := sum(7, 40, -8, 8) After shift coefficient0 = 47 Conditions: none Add all contributions to the same power. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore shift coefficient1 := sum(20, -8, 12) After shift coefficient1 = 24 Conditions: none Add all contributions to the same power. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore shift coefficient2 := sum(-2, 6) After shift coefficient2 = 4 Conditions: none Add all contributions to the same power. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore shift coefficient3 := sum(1) After shift coefficient3 = 1 Conditions: none Add all contributions to the same power. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · impliesBefore [47, 24, 4, 1] After canonical ascending coefficient list Conditions: none The result is the full coefficient identity for f(x+c);constant and zero polynomials are unchanged by translation. Step identifiers and dependenciesRule: polynomial_shift_coefficients · Uses: G001, S039 ## Reverse checks - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Recompute shift coefficient3 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute shift coefficient2 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute shift coefficient1 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute shift coefficient0 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute shift contribution3,3 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute shift scaled binomial3,3 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute shift contribution3,2 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute shift scaled binomial3,2 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Recompute shift contribution3,1 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute shift scaled binomial3,1 with exact rational arithmetic, not rounded decimals. - 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 shift contribution3,0 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute shift scaled binomial3,0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 shift contribution2,2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute shift scaled binomial2,2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute shift contribution2,1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute shift scaled binomial2,1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute shift contribution2,0 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute shift scaled binomial2,0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute shift contribution1,1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute shift scaled binomial1,1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute shift contribution1,0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute shift scaled binomial1,0 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute shift contribution0,0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute shift scaled binomial0,0 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 [47, 24, 4, 1] 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 343a33a72afe9fc67626327f6babc7f713b0069c3df242b5def0ed0772b03e1a - candidate record SHA-256: e55baf3e7e4f8e107a619b83ed821b860fad743e5c173ec7af9612681d1b46bd - semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a - question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r01-polynomial-0095.json)