Question 10013: Verifying a Difference-of-Squares Identity

Original question and givens

In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(12)² equal to the proposed coefficient vector [-144, 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−(12)² equal to the proposed coefficient vector [-144, 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

Step-by-step derivation

Q[x];ascending coefficients;input vectors may have trailing zeros

equality uses every normalized coefficient,not numerical sampling

original d=12;x^2-d^2

factors x-d and x+d;ascending vectors=[[-12, 1], [12, 1]]

power k=0;left=[-12, 1];right=[12, 1]

all index pairs i+j=k=[[0, 0]]

coefficient0 term0,0 := mul(-12, 12)

coefficient0 term0,0 = -144

coefficient0 complete sum := sum(-144)

coefficient0 complete sum = -144

power k=1;left=[-12, 1];right=[12, 1]

all index pairs i+j=k=[[0, 1], [1, 0]]

coefficient1 term0,1 := mul(-12, 1)

coefficient1 term0,1 = -12

coefficient1 term1,0 := mul(1, 12)

coefficient1 term1,0 = 12

coefficient1 complete sum := sum(-12, 12)

coefficient1 complete sum = 0

power k=2;left=[-12, 1];right=[12, 1]

all index pairs i+j=k=[[1, 1]]

coefficient2 term1,1 := mul(1, 1)

coefficient2 term1,1 = 1

coefficient2 complete sum := sum(1)

coefficient2 complete sum = 1

all collected product coefficients=[-144, 0, 1]

normalized complete product=[-144, 0, 1]

power k=0;actual coefficient=-144;candidate coefficient=-144

coefficient matches=Yes

power k=1;actual coefficient=0;candidate coefficient=0

coefficient matches=Yes

power k=2;actual coefficient=1;candidate coefficient=1

coefficient matches=Yes

raw proposed=[-144, 0, 1];normalized proposed=[-144, 0, 1];actual product=[-144, 0, 1]

coefficient identity=Yes

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.