Question 10614: Evaluate with Fixed Bivariate Coefficient Roles

Original question and givens

Use the declared real polynomial R²→R with coefficients=[1, 1, 3, 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=1, y=3. Give (coefficients,x,y,exact polynomial value).

• G001 — Use the declared real polynomial R²→R with coefficients=[1, 1, 3, 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=1, y=3. 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

Step-by-step derivation

original six coefficients=[1, 1, 3, 1, 0, -1]

declared scalar R2->R polynomial;finite audits keep coordinate/value roles

query point x squared := mul(1, 1)

query point x squared = 1

query point x times y := mul(1, 3)

query point x times y = 3

query point y squared := mul(3, 3)

query point y squared = 9

query point:x=1;y=3;coefficients=[1, 1, 3, 1, 0, -1]

fixed monomial values [1,x,y,x^2,xy,y^2]=[1, 1, 3, 1, 3, 9]

query point monomial term0 := mul(1, 1)

query point monomial term0 = 1

query point monomial term1 := mul(1, 1)

query point monomial term1 = 1

query point monomial term2 := mul(3, 3)

query point monomial term2 = 9

query point monomial term3 := mul(1, 1)

query point monomial term3 = 1

query point monomial term4 := mul(0, 3)

query point monomial term4 = 0

query point monomial term5 := mul(-1, 9)

query point monomial term5 = -9

query point complete polynomial sum := sum(1, 1, 9, 1, 0, -9)

query point complete polynomial sum = 3

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

([1, 1, 3, 1, 0, -1], 1, 3, 3)

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.