Complete Polynomial Coefficients and Ordered Values
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Univariate polynomials over the rationals · qf-density10-poly-shift-coefficients · candidate
Canonical question JSONL · English JSONL · Validation report