# Complete Polynomial Coefficients and Ordered Values

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261008-advance-20261010-r01-polynomial-foundations · 2026-10-08 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[0] and g=[0], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[0]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0001/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 2. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[2] and g=[-3], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0002/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 3. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[3, -2, 1] and g=[-3, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[0]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0003/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 4. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-9, 4, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-8, 2, 1, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-8, 2, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0004/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 5. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-8, 5, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-7, 3, 2]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-7, 3, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0005/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 6. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-7, 6, 1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-6, 4, 3, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-6, 4, 3, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0006/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 7. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-6, 7, -1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-5, 5, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-5, 5, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0007/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 8. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-5, 8, 0, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-4, 6, 2, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-4, 6, 2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0008/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 9. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-4, 9, 1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-3, 7, 3]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-3, 7, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0009/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 10. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-3, 10, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-2, 8, 1, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-2, 8, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0010/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 11. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-2, 11, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[-1, 9, 2]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [-1, 9, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0011/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 12. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[-1, 12, 1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[0, 10, 3, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [0, 10, 3, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0012/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 13. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[0, 13, -1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[1, 11, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [1, 11, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0013/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 14. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[1, 14, 0, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[2, 12, 2, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [2, 12, 2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0014/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 15. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[2, 15, 1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[3, 13, 3]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [3, 13, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0015/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 16. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[3, 16, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[4, 14, 1, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [4, 14, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0016/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 17. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[4, 17, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[5, 15, 2]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [5, 15, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0017/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 18. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[5, 18, 1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[6, 16, 3, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [6, 16, 3, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0018/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 19. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[6, 19, -1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[7, 17, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [7, 17, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0019/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 20. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[7, 20, 0, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[8, 18, 2, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [8, 18, 2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0020/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 21. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[8, 21, 1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[9, 19, 3]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [9, 19, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0021/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 22. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[9, 22, -1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[10, 20, 1, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [10, 20, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0022/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 23. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[10, 23, 0, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[11, 21, 2]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [11, 21, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0023/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 24. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[11, 24, 1, 2] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[12, 22, 3, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [12, 22, 3, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0024/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 25. Complete Polynomial Sum Coefficients

Integer polynomials have ascending coefficients f=[12, 25, -1, 1] and g=[1, -2, 2, -1], constant term first. Give the complete ascending coefficient list for f+g.

View answer and explanation[13, 23, 1]

Add corresponding powers, remove trailing zero coefficients, and retain [0] for the zero polynomial; one-point equality is not a polynomial identity. The exact result, retaining the requested order, is [13, 23, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0025/)

Univariate polynomials over the rationals · qf-density10-poly-add-coefficients · candidate





## 26. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[0] and g=[1, -1], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[0]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0026/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 27. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[2] and g=[-3], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-6]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-6].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0027/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 28. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[1, 1] and g=[1, -1], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[1, 0, -1]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [1, 0, -1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0028/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 29. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-9, 4, -3, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-18, 17, -19, -9, 4, -5, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-18, 17, -19, -9, 4, -5, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0029/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 30. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-8, 5, -4, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-16, 18, -21, -5, 5, -7, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-16, 18, -21, -5, 5, -7, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0030/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 31. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-7, 6, -5, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-14, 19, -23, -1, 6, -9, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-14, 19, -23, -1, 6, -9, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0031/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 32. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-6, 7, -6, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-12, 20, -25, 3, 7, -11, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-12, 20, -25, 3, 7, -11, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0032/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 33. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-5, 8, -7, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-10, 21, -27, 7, 8, -13, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-10, 21, -27, 7, 8, -13, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0033/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 34. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-4, 9, -8, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-8, 22, -29, 11, 9, -15, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-8, 22, -29, 11, 9, -15, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0034/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 35. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-3, 10, -9, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-6, 23, -31, 15, 10, -17, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-6, 23, -31, 15, 10, -17, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0035/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 36. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-2, 11, -10, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-4, 24, -33, 19, 11, -19, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-4, 24, -33, 19, 11, -19, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0036/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 37. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[-1, 12, -11, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[-2, 25, -35, 23, 12, -21, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [-2, 25, -35, 23, 12, -21, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0037/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 38. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[0, 13, -12, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[0, 26, -37, 27, 13, -23, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [0, 26, -37, 27, 13, -23, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0038/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 39. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[1, 14, -13, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[2, 27, -39, 31, 14, -25, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [2, 27, -39, 31, 14, -25, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0039/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 40. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[2, 15, -14, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[4, 28, -41, 35, 15, -27, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [4, 28, -41, 35, 15, -27, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0040/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 41. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[3, 16, -15, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[6, 29, -43, 39, 16, -29, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [6, 29, -43, 39, 16, -29, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0041/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 42. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[4, 17, -16, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[8, 30, -45, 43, 17, -31, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [8, 30, -45, 43, 17, -31, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0042/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 43. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[5, 18, -17, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[10, 31, -47, 47, 18, -33, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [10, 31, -47, 47, 18, -33, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0043/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 44. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[6, 19, -18, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[12, 32, -49, 51, 19, -35, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [12, 32, -49, 51, 19, -35, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0044/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 45. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[7, 20, -19, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[14, 33, -51, 55, 20, -37, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [14, 33, -51, 55, 20, -37, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0045/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 46. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[8, 21, -20, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[16, 34, -53, 59, 21, -39, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [16, 34, -53, 59, 21, -39, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0046/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 47. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[9, 22, -21, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[18, 35, -55, 63, 22, -41, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [18, 35, -55, 63, 22, -41, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0047/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 48. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[10, 23, -22, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[20, 36, -57, 67, 23, -43, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [20, 36, -57, 67, 23, -43, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0048/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 49. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[11, 24, -23, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[22, 37, -59, 71, 24, -45, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [22, 37, -59, 71, 24, -45, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0049/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 50. Complete Polynomial Product Coefficients

Integer polynomials have ascending coefficients f=[12, 25, -24, 1] and g=[2, -1, 1, 2], constant term first. Give the complete ascending coefficients of fg, potentially through degree 6, without truncation.

View answer and explanation[24, 38, -61, 75, 25, -47, 2]

Combine all product powers before trimming trailing zeros; a zero factor gives [0]. The exact result, retaining the requested order, is [24, 38, -61, 75, 25, -47, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0050/)

Univariate polynomials over the rationals · qf-density10-poly-product-coefficients · candidate





## 51. Ordered Polynomial Values at Two Points

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

View answer and explanation(0, 0)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0051/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 52. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-3], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.

View answer and explanation(-3, -3)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-3, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0052/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 53. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-10, 2, -1, 2], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.

View answer and explanation(-34, 41)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-34, 41).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0053/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 54. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-9, 2, -1, 0], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.

View answer and explanation(-17, -17)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-17, -17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0054/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 55. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-8, 2, -1, 1], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.

View answer and explanation(-24, -6)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-24, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0055/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 56. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-7, 2, -1, 2], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.

View answer and explanation(-31, 9)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-31, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0056/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 57. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-6, 2, -1, 0], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.

View answer and explanation(-14, -9)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-14, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0057/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 58. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-5, 2, -1, 1], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.

View answer and explanation(-21, 51)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-21, 51).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0058/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 59. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-4, 2, -1, 2], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.

View answer and explanation(-28, -1)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-28, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0059/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 60. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-3, 2, -1, 0], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.

View answer and explanation(-11, -3)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-11, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0060/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 61. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-2, 2, -1, 1], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.

View answer and explanation(-18, 22)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-18, 22).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0061/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 62. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[-1, 2, -1, 2], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.

View answer and explanation(-25, 119)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-25, 119).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0062/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 63. Ordered Polynomial Values at Two Points

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.

View answer and explanation(-8, 1)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-8, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0063/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 64. Ordered Polynomial Values at Two Points

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.

View answer and explanation(-15, 9)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-15, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0064/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 65. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[2, 2, -1, 2], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.

View answer and explanation(-22, 53)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-22, 53).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0065/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 66. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[3, 2, -1, 0], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.

View answer and explanation(-5, -5)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-5, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0066/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 67. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[4, 2, -1, 1], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.

View answer and explanation(-12, 6)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-12, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0067/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 68. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[5, 2, -1, 2], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.

View answer and explanation(-19, 21)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-19, 21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0068/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 69. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[6, 2, -1, 0], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.

View answer and explanation(-2, 3)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-2, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0069/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 70. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[7, 2, -1, 1], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.

View answer and explanation(-9, 63)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-9, 63).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0070/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 71. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[8, 2, -1, 2], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.

View answer and explanation(-16, 11)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-16, 11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0071/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 72. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[9, 2, -1, 0], constant term first. For a=-2, b=2, give (f(a), f(b)) as integers.

View answer and explanation(1, 9)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (1, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0072/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 73. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[10, 2, -1, 1], constant term first. For a=-2, b=3, give (f(a), f(b)) as integers.

View answer and explanation(-6, 34)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-6, 34).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0073/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 74. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[11, 2, -1, 2], constant term first. For a=-2, b=4, give (f(a), f(b)) as integers.

View answer and explanation(-13, 131)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (-13, 131).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0074/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 75. Ordered Polynomial Values at Two Points

Integer polynomials have ascending coefficients f=[12, 2, -1, 0], constant term first. For a=-2, b=1, give (f(a), f(b)) as integers.

View answer and explanation(4, 13)

Evaluate at each specified point in order; equal results at two points do not establish a polynomial identity. The exact result, retaining the requested order, is (4, 13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0075/)

Univariate polynomials over the rationals · qf-density10-poly-two-point-values · candidate





## 76. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[0], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[0]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0076/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 77. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-4], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-4]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0077/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 78. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-10, 3, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-10, 3, -2, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-10, 3, -2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0078/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 79. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-9, 4, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-6, 3, 1, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-6, 3, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0079/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 80. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-8, 5, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[2, 9, 4, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [2, 9, 4, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0080/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 81. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-7, 6, -2, 1], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-35, 26, -8, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-35, 26, -8, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0081/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 82. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-6, 7, -2, 1], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-16, 14, -5, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-16, 14, -5, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0082/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 83. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-5, 8, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-5, 8, -2, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-5, 8, -2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0083/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 84. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-4, 9, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[4, 8, 1, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [4, 8, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0084/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 85. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-3, 10, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[17, 14, 4, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [17, 14, 4, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0085/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 86. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-2, 11, -2, 1], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-40, 31, -8, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-40, 31, -8, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0086/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 87. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[-1, 12, -2, 1], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-16, 19, -5, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-16, 19, -5, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0087/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 88. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[0, 13, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[0, 13, -2, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [0, 13, -2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0088/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 89. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[1, 14, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[14, 13, 1, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [14, 13, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0089/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 90. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[2, 15, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[32, 19, 4, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [32, 19, 4, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0090/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 91. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[3, 16, -2, 1], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-45, 36, -8, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-45, 36, -8, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0091/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 92. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[4, 17, -2, 1], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-16, 24, -5, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-16, 24, -5, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0092/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 93. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[5, 18, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[5, 18, -2, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [5, 18, -2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0093/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 94. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[6, 19, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[24, 18, 1, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [24, 18, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0094/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 95. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[7, 20, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[47, 24, 4, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [47, 24, 4, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0095/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 96. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[8, 21, -2, 1], constant term first. For c=-2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-50, 41, -8, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-50, 41, -8, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0096/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 97. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[9, 22, -2, 1], constant term first. For c=-1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[-16, 29, -5, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [-16, 29, -5, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0097/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 98. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[10, 23, -2, 1], constant term first. For c=0, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[10, 23, -2, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [10, 23, -2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0098/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 99. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[11, 24, -2, 1], constant term first. For c=1, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[34, 23, 1, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [34, 23, 1, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0099/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate





## 100. Complete Polynomial Translation Coefficients

Integer polynomials have ascending coefficients f=[12, 25, -2, 1], constant term first. For c=2, give the complete ascending coefficients of f(x+c), not merely a value at one point.

View answer and explanation[62, 29, 4, 1]

Expand the coefficient recurrence P←(x+c)P+a through every power; a constant is unchanged by translation. The exact result, retaining the requested order, is [62, 29, 4, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r01-polynomial-0100/)

Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate



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