# Polynomial Factorization Foundations

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

batch-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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0025/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0026/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0027/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0028/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0029/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0030/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0031/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0032/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0033/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0034/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0035/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0036/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0037/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0038/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0039/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0040/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0041/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0042/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0043/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0044/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0045/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0046/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0047/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0048/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0049/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0051/)

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 explanation7

Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0054/)

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 explanation0

Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0055/)

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 explanation0

Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0056/)

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 explanation8

Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0057/)

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 explanation7

Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0060/)

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 explanation0

Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0061/)

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 explanation0

Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0062/)

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 explanation98

Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=98.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0063/)

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 explanation7

Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0066/)

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 explanation0

Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0067/)

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 explanation0

Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0068/)

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 explanation260

Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=260.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0069/)

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 explanation7

Sum products of coefficients whose indices add to 1. The exact product coefficient is a_1=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0072/)

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 explanation0

Sum products of coefficients whose indices add to 4. The exact product coefficient is a_4=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0073/)

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 explanation0

Sum products of coefficients whose indices add to 5. The exact product coefficient is a_5=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0074/)

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 explanation494

Sum products of coefficients whose indices add to 0. The exact product coefficient is a_0=494.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0075/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0076/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0077/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0078/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0079/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0080/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0081/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0082/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0083/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0084/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0085/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0086/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0087/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0088/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0089/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0090/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0091/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0092/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0093/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0094/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0095/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0096/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0097/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0098/)

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 explanationNo

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0099/)

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 explanationYes

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](https://ocme.evemisslab.com/en/questions/q-20261003-r03-factorization-0100/)

Polynomial factorization over the rationals · qf-factorization-difference-squares · candidate



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