# Question 10021: Verifying a Difference-of-Squares Identity [Return to catalogue page 51](https://ocme.evemisslab.com/en/questions/page/51/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-r03-factorization-foundations/#q-20261003-r03-factorization-0096) 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. R0019-10021 · q-20261003-r03-factorization-0096 ## Original question and givens In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(20)² equal to the proposed coefficient vector [-400, 0, 1]? Answer Yes or No. - G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(20)² equal to the proposed coefficient vector [-400, 0, 1]? Answer Yes or No. ## 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-factorization](https://ocme.evemisslab.com/en/mko/mko-factorization/) ## Step-by-step derivation S001 · impliesBefore Q[x];ascending coefficients;input vectors may have trailing zeros After equality uses every normalized coefficient,not numerical sampling Conditions: none Preserve the original factor order and vectors.Normalization removes only trailing high-power zeros;do not reorder source factors by semantic identity or infer polynomial equality from one point. Step identifiers and dependenciesRule: factorization_coefficient_identity_scope · Uses: G001 S002 · impliesBefore original d=20;x^2-d^2 After factors x-d and x+d;ascending vectors=[[-20, 1], [20, 1]] Conditions: none Expand(x−d)(x+d)=x²+dx−dx−d²,retaining the original sign of d.The same expansion applies for negative d or d=0. Step identifiers and dependenciesRule: difference_of_squares_actual_signed_parameter · Uses: G001, S001 S003 · impliesBefore power k=0;left=[-20, 1];right=[20, 1] After all index pairs i+j=k=[[0, 0]] Conditions: none Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication. Step identifiers and dependenciesRule: product_coefficient_complete_index_pairs · Uses: G001, S002 S004 · calculationBefore coefficient0 term0,0 := mul(-20, 20) After coefficient0 term0,0 = -400 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore coefficient0 complete sum := sum(-400) After coefficient0 complete sum = -400 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore power k=1;left=[-20, 1];right=[20, 1] After all index pairs i+j=k=[[0, 1], [1, 0]] Conditions: none Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication. Step identifiers and dependenciesRule: product_coefficient_complete_index_pairs · Uses: G001, S005 S007 · calculationBefore coefficient1 term0,1 := mul(-20, 1) After coefficient1 term0,1 = -20 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore coefficient1 term1,0 := mul(1, 20) After coefficient1 term1,0 = 20 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore coefficient1 complete sum := sum(-20, 20) After coefficient1 complete sum = 0 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore power k=2;left=[-20, 1];right=[20, 1] After all index pairs i+j=k=[[1, 1]] Conditions: none Ascending-vector indices are powers.Collect every valid pair with i+j=k;polynomial multiplication is not entrywise multiplication. Step identifiers and dependenciesRule: product_coefficient_complete_index_pairs · Uses: G001, S009 S011 · calculationBefore coefficient2 term1,1 := mul(1, 1) After coefficient2 term1,1 = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore coefficient2 complete sum := sum(1) After coefficient2 complete sum = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore all collected product coefficients=[-400, 0, 1] After normalized complete product=[-400, 0, 1] Conditions: none All finitely many coefficients were obtained.Remove trailing high-power zeros and display zero as[0].This is coefficient identity,not agreement at a few sample points. Step identifiers and dependenciesRule: complete_product_not_sampled_values · Uses: G001, S012 S014 · impliesBefore power k=0;actual coefficient=-400;candidate coefficient=-400 After coefficient matches=Yes Conditions: none Treat absent higher coefficients as0 and compare through the larger degree.The supplied candidate is not overwritten by the correct product. Step identifiers and dependenciesRule: every_identity_coefficient_comparison · Uses: G001, S013 S015 · impliesBefore power k=1;actual coefficient=0;candidate coefficient=0 After coefficient matches=Yes Conditions: none Treat absent higher coefficients as0 and compare through the larger degree.The supplied candidate is not overwritten by the correct product. Step identifiers and dependenciesRule: every_identity_coefficient_comparison · Uses: G001, S014 S016 · impliesBefore power k=2;actual coefficient=1;candidate coefficient=1 After coefficient matches=Yes Conditions: none Treat absent higher coefficients as0 and compare through the larger degree.The supplied candidate is not overwritten by the correct product. Step identifiers and dependenciesRule: every_identity_coefficient_comparison · Uses: G001, S015 S017 · impliesBefore raw proposed=[-400, 0, 1];normalized proposed=[-400, 0, 1];actual product=[-400, 0, 1] After coefficient identity=Yes Conditions: none Identity requires equality of all coefficients,including the both-zero case.Constants and trailing-zero representations do not create a different polynomial. Step identifiers and dependenciesRule: factorization_final_full_identity_gate · Uses: G001, S016 ## Reverse checks - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — 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 coefficient2 complete sum with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute coefficient2 term1,1 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 coefficient1 complete sum with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute coefficient1 term1,0 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute coefficient1 term0,1 with exact rational arithmetic, not rounded decimals. - 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 coefficient0 complete sum with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute coefficient0 term0,0 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 Yes 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: 844f947dc87d0c24091554aae292e1742e9a023028fa34c5da0bc8e9d6232dad - candidate record SHA-256: f32fb41d706b3a33ad63f559a79f2c68f4ebe64ae58afa2b5d154d63e58b8958 - semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5 - question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-r03-factorization-0096.json)