Polynomial Factorization Foundations
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batch-20261003-r03-factorization-foundations · 2026-10-03 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [0] and [7, 1] and return the normalized complete coefficient vector.
View answer and explanation
[0]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [0]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
2. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-11, 1] and [3, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-33, -8, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-33, -8, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
3. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-10, 1] and [4, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-40, -6, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-40, -6, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
4. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-9, 1] and [5, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-45, -4, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-45, -4, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
5. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-8, 1] and [6, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-48, -2, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-48, -2, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
6. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-7, 1] and [7, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-49, 0, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-49, 0, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
7. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-6, 1] and [8, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-48, 2, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-48, 2, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
8. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-5, 1] and [9, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-45, 4, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-45, 4, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
9. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-4, 1] and [10, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-40, 6, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-40, 6, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
10. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-3, 1] and [11, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-33, 8, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-33, 8, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
11. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-2, 1] and [12, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-24, 10, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-24, 10, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
12. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [-1, 1] and [13, 1] and return the normalized complete coefficient vector.
View answer and explanation
[-13, 12, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [-13, 12, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
13. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [0, 1] and [14, 1] and return the normalized complete coefficient vector.
View answer and explanation
[0, 14, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [0, 14, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
14. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [1, 1] and [15, 1] and return the normalized complete coefficient vector.
View answer and explanation
[15, 16, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [15, 16, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
15. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [2, 1] and [16, 1] and return the normalized complete coefficient vector.
View answer and explanation
[32, 18, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [32, 18, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
16. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [3, 1] and [17, 1] and return the normalized complete coefficient vector.
View answer and explanation
[51, 20, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [51, 20, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
17. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [4, 1] and [18, 1] and return the normalized complete coefficient vector.
View answer and explanation
[72, 22, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [72, 22, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
18. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [5, 1] and [19, 1] and return the normalized complete coefficient vector.
View answer and explanation
[95, 24, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [95, 24, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
19. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [6, 1] and [20, 1] and return the normalized complete coefficient vector.
View answer and explanation
[120, 26, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [120, 26, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
20. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [7, 1] and [21, 1] and return the normalized complete coefficient vector.
View answer and explanation
[147, 28, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [147, 28, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
21. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [8, 1] and [22, 1] and return the normalized complete coefficient vector.
View answer and explanation
[176, 30, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [176, 30, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
22. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [9, 1] and [23, 1] and return the normalized complete coefficient vector.
View answer and explanation
[207, 32, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [207, 32, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
23. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [10, 1] and [24, 1] and return the normalized complete coefficient vector.
View answer and explanation
[240, 34, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [240, 34, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
24. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [11, 1] and [25, 1] and return the normalized complete coefficient vector.
View answer and explanation
[275, 36, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [275, 36, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
25. Complete Coefficients of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Multiply the factors [12, 1] and [26, 1] and return the normalized complete coefficient vector.
View answer and explanation
[312, 38, 1]
Multiply term by term and collect coefficients with the same exponent sum. The complete product is [312, 38, 1]; the identity is determined by all coefficients rather than sampled values.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product · candidate
26. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [0] and [7, 1] equal to the proposed polynomial [0]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
27. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-11, 1] and [3, 1] equal to the proposed polynomial [-32, -8, 1]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
28. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-10, 1] and [4, 1] equal to the proposed polynomial [-40, -6, 2]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
29. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-9, 1] and [5, 1] equal to the proposed polynomial [-45, -4, 1]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
30. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-8, 1] and [6, 1] equal to the proposed polynomial [-47, -2, 1]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
31. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-7, 1] and [7, 1] equal to the proposed polynomial [-49, 0, 2]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
32. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-6, 1] and [8, 1] equal to the proposed polynomial [-48, 2, 1]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
33. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-5, 1] and [9, 1] equal to the proposed polynomial [-44, 4, 1]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
34. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-4, 1] and [10, 1] equal to the proposed polynomial [-40, 6, 2]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
35. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-3, 1] and [11, 1] equal to the proposed polynomial [-33, 8, 1]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
36. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-2, 1] and [12, 1] equal to the proposed polynomial [-23, 10, 1]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
37. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [-1, 1] and [13, 1] equal to the proposed polynomial [-13, 12, 2]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
38. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [0, 1] and [14, 1] equal to the proposed polynomial [0, 14, 1]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
39. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [1, 1] and [15, 1] equal to the proposed polynomial [16, 16, 1]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
40. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [2, 1] and [16, 1] equal to the proposed polynomial [32, 18, 2]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
41. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [3, 1] and [17, 1] equal to the proposed polynomial [51, 20, 1]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
42. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [4, 1] and [18, 1] equal to the proposed polynomial [73, 22, 1]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
43. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [5, 1] and [19, 1] equal to the proposed polynomial [95, 24, 2]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
44. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [6, 1] and [20, 1] equal to the proposed polynomial [120, 26, 1]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
45. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [7, 1] and [21, 1] equal to the proposed polynomial [148, 28, 1]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
46. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [8, 1] and [22, 1] equal to the proposed polynomial [176, 30, 2]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
47. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [9, 1] and [23, 1] equal to the proposed polynomial [207, 32, 1]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
48. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [10, 1] and [24, 1] equal to the proposed polynomial [241, 34, 1]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
49. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [11, 1] and [25, 1] equal to the proposed polynomial [275, 36, 2]? Answer Yes or No.
View answer and explanation
No
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives No; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
50. Verifying a Factorization by Coefficients
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the product of [12, 1] and [26, 1] equal to the proposed polynomial [312, 38, 1]? Answer Yes or No.
View answer and explanation
Yes
Expand the factors and compare every coefficient, ignoring trailing zeros. The coefficient identity gives Yes; this is not an inference from numerical sampling.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-identity · candidate
51. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [-5, 1] and [2, -1].
View answer and explanation
-10
Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=-10.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
52. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [-4, 1, 1] and [3, -1].
View answer and explanation
7
Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
53. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^2 in the product of [-3, 1] and [4, -1].
View answer and explanation
-1
Sum products of coefficients whose indices add to 2. The exact product coefficient is a_2=-1.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
54. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^3 in the product of [-2, 1, 1] and [5, -1].
View answer and explanation
-1
Sum products of coefficients whose indices add to 3. The exact product coefficient is a_3=-1.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
55. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^4 in the product of [-1, 1] and [6, -1].
View answer and explanation
0
Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
56. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [0, 1, 1] and [7, -1].
View answer and explanation
0
Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
57. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [1, 1] and [8, -1].
View answer and explanation
8
Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=8.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
58. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [2, 1, 1] and [9, -1].
View answer and explanation
7
Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
59. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^2 in the product of [3, 1] and [10, -1].
View answer and explanation
-1
Sum products of coefficients whose indices add to 2. The exact product coefficient is a_2=-1.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
60. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^3 in the product of [4, 1, 1] and [11, -1].
View answer and explanation
-1
Sum products of coefficients whose indices add to 3. The exact product coefficient is a_3=-1.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
61. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^4 in the product of [5, 1] and [12, -1].
View answer and explanation
0
Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
62. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [6, 1, 1] and [13, -1].
View answer and explanation
0
Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
63. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [7, 1] and [14, -1].
View answer and explanation
98
Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=98.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
64. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [8, 1, 1] and [15, -1].
View answer and explanation
7
Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
65. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^2 in the product of [9, 1] and [16, -1].
View answer and explanation
-1
Sum products of coefficients whose indices add to 2. The exact product coefficient is a_2=-1.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
66. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^3 in the product of [10, 1, 1] and [17, -1].
View answer and explanation
-1
Sum products of coefficients whose indices add to 3. The exact product coefficient is a_3=-1.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
67. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^4 in the product of [11, 1] and [18, -1].
View answer and explanation
0
Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
68. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [12, 1, 1] and [19, -1].
View answer and explanation
0
Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
69. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [13, 1] and [20, -1].
View answer and explanation
260
Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=260.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
70. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^1 in the product of [14, 1, 1] and [21, -1].
View answer and explanation
7
Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
71. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^2 in the product of [15, 1] and [22, -1].
View answer and explanation
-1
Sum products of coefficients whose indices add to 2. The exact product coefficient is a_2=-1.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
72. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^3 in the product of [16, 1, 1] and [23, -1].
View answer and explanation
-1
Sum products of coefficients whose indices add to 3. The exact product coefficient is a_3=-1.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
73. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^4 in the product of [17, 1] and [24, -1].
View answer and explanation
0
Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
74. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^5 in the product of [18, 1, 1] and [25, -1].
View answer and explanation
0
Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
75. Specified Coefficient of a Factor Product
In Q[x], coefficient vectors are ordered from lowest to highest power. Find the coefficient of x^0 in the product of [19, 1] and [26, -1].
View answer and explanation
494
Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=494.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-product-coefficient · candidate
76. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(0)² equal to the proposed coefficient vector [0, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(0)² is [0, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
77. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(1)² equal to the proposed coefficient vector [0, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(1)² is [-1, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
78. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(2)² equal to the proposed coefficient vector [-4, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(2)² is [-4, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
79. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(3)² equal to the proposed coefficient vector [-8, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(3)² is [-9, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
80. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(4)² equal to the proposed coefficient vector [-16, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(4)² is [-16, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
81. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(5)² equal to the proposed coefficient vector [-24, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(5)² is [-25, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
82. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(6)² equal to the proposed coefficient vector [-36, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(6)² is [-36, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
83. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(7)² equal to the proposed coefficient vector [-48, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(7)² is [-49, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
84. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(8)² equal to the proposed coefficient vector [-64, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(8)² is [-64, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
85. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(9)² equal to the proposed coefficient vector [-80, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(9)² is [-81, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
86. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(10)² equal to the proposed coefficient vector [-100, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(10)² is [-100, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
87. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(11)² equal to the proposed coefficient vector [-120, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(11)² is [-121, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
88. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(12)² equal to the proposed coefficient vector [-144, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(12)² is [-144, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
89. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(13)² equal to the proposed coefficient vector [-168, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(13)² is [-169, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
90. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(14)² equal to the proposed coefficient vector [-196, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(14)² is [-196, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
91. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(15)² equal to the proposed coefficient vector [-224, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(15)² is [-225, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
92. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(16)² equal to the proposed coefficient vector [-256, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(16)² is [-256, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
93. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(17)² equal to the proposed coefficient vector [-288, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(17)² is [-289, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
94. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(18)² equal to the proposed coefficient vector [-324, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(18)² is [-324, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
95. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(19)² equal to the proposed coefficient vector [-360, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(19)² is [-361, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
96. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(20)² equal to the proposed coefficient vector [-400, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(20)² is [-400, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
97. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(21)² equal to the proposed coefficient vector [-440, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(21)² is [-441, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
98. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(22)² equal to the proposed coefficient vector [-484, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(22)² is [-484, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
99. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(23)² equal to the proposed coefficient vector [-528, 0, 1]? Answer Yes or No.
View answer and explanation
No
The full coefficient vector of x²−(23)² is [-529, 0, 1]. Comparing all coefficients after removing trailing zeros gives No.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
100. Verifying a Difference-of-Squares Identity
In Q[x], coefficient vectors are ordered from lowest to highest power. Is the difference of squares x²−(24)² equal to the proposed coefficient vector [-576, 0, 1]? Answer Yes or No.
View answer and explanation
Yes
The full coefficient vector of x²−(24)² is [-576, 0, 1]. Comparing all coefficients after removing trailing zeros gives Yes.
Reviewed step-by-step derivation
Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate
Canonical question JSONL · English JSONL · Validation report