# Question 10623: Evaluate with Fixed Bivariate Coefficient Roles [Return to catalogue page 54](https://ocme.evemisslab.com/en/questions/page/54/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261023-r07-multivariable-real-function-foundations/#q-20261013-advance-20261023-r07-multivariable-real-function-0023) 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. R0020-10623 · q-20261013-advance-20261023-r07-multivariable-real-function-0023 ## Original question and givens Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=-2. Give (coefficients,x,y,exact polynomial value). - G001 — Use the declared real polynomial R²→R with coefficients=[10, 0, -2, 1, 0, -1] in the fixed order [1,x,y,x²,x*y,y²]. Keep the two coordinate roles distinct. These explicit slices establish no global continuity, differentiability or vector-valued-function claim. x=0, y=-2. Give (coefficients,x,y,exact polynomial value). ## 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-multivariable-real-function](https://ocme.evemisslab.com/en/mko/mko-multivariable-real-function/) ## Step-by-step derivation S001 · impliesBefore original six coefficients=[10, 0, -2, 1, 0, -1] After declared scalar R2->R polynomial;finite audits keep coordinate/value roles Conditions: none Evaluate the declared formula exactly.A finite point match is not a new general formal proof of global equivalence,continuity or differentiability. Step identifiers and dependenciesRule: bivariate_explicit_slice_scope · Uses: G001 S002 · calculationBefore query point x squared := mul(0, 0) After query point x squared = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore query point x times y := mul(0, -2) After query point x times y = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore query point y squared := mul(-2, -2) After query point y squared = 4 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore query point:x=0;y=-2;coefficients=[10, 0, -2, 1, 0, -1] After fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 0, -2, 0, 0, 4] Conditions: none The six coefficients correspond to[1,x,y,x²,xy,y²] in that order.Do not exchange x and y or reinterpret the scalar function as vector-valued. Step identifiers and dependenciesRule: bivariate_fixed_coordinate_monomial_roles · Uses: G001, S004 S006 · calculationBefore query point monomial term0 := mul(10, 1) After query point monomial term0 = 10 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore query point monomial term1 := mul(0, 0) After query point monomial term1 = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore query point monomial term2 := mul(-2, -2) After query point monomial term2 = 4 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore query point monomial term3 := mul(1, 0) After query point monomial term3 = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore query point monomial term4 := mul(0, 0) After query point monomial term4 = 0 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore query point monomial term5 := mul(-1, 4) After query point monomial term5 = -4 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore query point complete polynomial sum := sum(10, 0, 4, 0, 0, -4) After query point complete polynomial sum = 10 Conditions: none Compute exactly with this step's original roles and operands,without floating approximation or reordering source data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 ## Reverse checks - S012 · premise_and_result_audit — Recompute query point complete polynomial sum with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute query point monomial term5 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute query point monomial term4 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute query point monomial term3 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute query point monomial term2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute query point monomial term1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute query point monomial term0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute query point y squared with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute query point x times y with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute query point x squared 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 ([10, 0, -2, 1, 0, -1], 0, -2, 10) 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: 0e9bab6597b7eb137a8e3046bb717734e471f15ae129e05669ea637b0632515a - candidate record SHA-256: 754a8b7d45782ea5ec5a67f839465ab8f5b1ae6df406816b802d59c6462b1e54 - semantic review receipt SHA-256: b4c47938fd38085d44967226cfa8337d61334455472e7307c60d3a488cef598f - question-first witness SHA-256: 3c23baa97a5933bc07b44f2257017ef4e845bf1429bd6f5381410681c43b97e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r07-multivariable-real-function-0023.json)