Question 4288: Ordered Polynomial Values at Two Points

Original question and givens

Integer polynomials have ascending coefficients f=[0, 2, -1, 0], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.

• G001 — Integer polynomials have ascending coefficients f=[0, 2, -1, 0], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.

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-polynomial

Step-by-step derivation

f ascending coefficients=[0, 2, -1, 0]

coefficient k multiplies x^k;keep all required degrees

[0, 2, -1, 0]

Horner at a=-2

a multiply before degree1 := mul(-1, -2)

a multiply before degree1 = 2

a add coefficient1 := sum(2, 2)

a add coefficient1 = 4

a multiply before degree0 := mul(4, -2)

a multiply before degree0 = -8

a add coefficient0 := sum(-8, 0)

a add coefficient0 = -8

[0, 2, -1, 0]

Horner at b=1

b multiply before degree1 := mul(-1, 1)

b multiply before degree1 = -1

b add coefficient1 := sum(-1, 2)

b add coefficient1 = 1

b multiply before degree0 := mul(1, 1)

b multiply before degree0 = 1

b add coefficient0 := sum(1, 0)

b add coefficient0 = 1

f(a)=-8;f(b)=1

retain a,b order;two equal values do not make f constant

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute b add coefficient0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute b multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute b add coefficient1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute b multiply before degree1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute a add coefficient0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute a multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute a add coefficient1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute a multiply before degree1 with exact rational arithmetic, not rounded decimals.
• 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

(-8, 1)

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.