Question 10034: Verifying a Factorization by Coefficients

Original question and givens

In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-4, 1] and [10, 1] equal to the proposed polynomial [-40, 6, 2]? Answer Yes or No.

• G001 — In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-4, 1] and [10, 1] equal to the proposed polynomial [-40, 6, 2]? 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

raw left=[-4, 1];raw right=[10, 1]

normalized left=[-4, 1];normalized right=[10, 1]

power k=0;left=[-4, 1];right=[10, 1]

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

coefficient0 term0,0 := mul(-4, 10)

coefficient0 term0,0 = -40

coefficient0 complete sum := sum(-40)

coefficient0 complete sum = -40

power k=1;left=[-4, 1];right=[10, 1]

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

coefficient1 term0,1 := mul(-4, 1)

coefficient1 term0,1 = -4

coefficient1 term1,0 := mul(1, 10)

coefficient1 term1,0 = 10

coefficient1 complete sum := sum(-4, 10)

coefficient1 complete sum = 6

power k=2;left=[-4, 1];right=[10, 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=[-40, 6, 1]

normalized complete product=[-40, 6, 1]

power k=0;actual coefficient=-40;candidate coefficient=-40

coefficient matches=Yes

power k=1;actual coefficient=6;candidate coefficient=6

coefficient matches=Yes

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

coefficient matches=No

raw proposed=[-40, 6, 2];normalized proposed=[-40, 6, 2];actual product=[-40, 6, 1]

coefficient identity=No

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

No

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.