Question 4289: Ordered Polynomial Values at Two Points

Original question and givens

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

• G001 — Integer polynomials have ascending coefficients f=[1, 2, -1, 1], constant term first. For a=-2, b=2, 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=[1, 2, -1, 1]

coefficient k multiplies x^k;keep all required degrees

[1, 2, -1, 1]

Horner at a=-2

a multiply before degree2 := mul(1, -2)

a multiply before degree2 = -2

a add coefficient2 := sum(-2, -1)

a add coefficient2 = -3

a multiply before degree1 := mul(-3, -2)

a multiply before degree1 = 6

a add coefficient1 := sum(6, 2)

a add coefficient1 = 8

a multiply before degree0 := mul(8, -2)

a multiply before degree0 = -16

a add coefficient0 := sum(-16, 1)

a add coefficient0 = -15

[1, 2, -1, 1]

Horner at b=2

b multiply before degree2 := mul(1, 2)

b multiply before degree2 = 2

b add coefficient2 := sum(2, -1)

b add coefficient2 = 1

b multiply before degree1 := mul(1, 2)

b multiply before degree1 = 2

b add coefficient1 := sum(2, 2)

b add coefficient1 = 4

b multiply before degree0 := mul(4, 2)

b multiply before degree0 = 8

b add coefficient0 := sum(8, 1)

b add coefficient0 = 9

f(a)=-15;f(b)=9

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

Reverse checks

• 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 — Recompute b add coefficient0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute b multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute b add coefficient1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute b multiply before degree1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute b add coefficient2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute b multiply before degree2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute a add coefficient0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute a multiply before degree0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute a add coefficient1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute a multiply before degree1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute a add coefficient2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute a multiply before degree2 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

(-15, 9)

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.