# Univariate Polynomial 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-r02-polynomial-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. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [0]. Compute p(-12) exactly.

View answer and explanation0

Multiply each coefficient by the corresponding natural-number power and add: p(-12)=0. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0001/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 2. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5]. Compute p(-11) exactly.

View answer and explanation5

Multiply each coefficient by the corresponding natural-number power and add: p(-11)=5. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0002/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 3. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [0, 1]. Compute p(-10) exactly.

View answer and explanation-10

Multiply each coefficient by the corresponding natural-number power and add: p(-10)=-10. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0003/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 4. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-9, 0, -1]. Compute p(-9) exactly.

View answer and explanation-90

Multiply each coefficient by the corresponding natural-number power and add: p(-9)=-90. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0004/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 5. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-8, 1]. Compute p(-8) exactly.

View answer and explanation-16

Multiply each coefficient by the corresponding natural-number power and add: p(-8)=-16. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0005/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 6. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-7, 2, 1]. Compute p(-7) exactly.

View answer and explanation28

Multiply each coefficient by the corresponding natural-number power and add: p(-7)=28. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0006/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 7. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-6, 3, -1]. Compute p(-6) exactly.

View answer and explanation-60

Multiply each coefficient by the corresponding natural-number power and add: p(-6)=-60. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0007/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 8. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-5, -3]. Compute p(-5) exactly.

View answer and explanation10

Multiply each coefficient by the corresponding natural-number power and add: p(-5)=10. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0008/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 9. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-4, -2, 1]. Compute p(-4) exactly.

View answer and explanation20

Multiply each coefficient by the corresponding natural-number power and add: p(-4)=20. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0009/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 10. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-3, -1, -1]. Compute p(-3) exactly.

View answer and explanation-9

Multiply each coefficient by the corresponding natural-number power and add: p(-3)=-9. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0010/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 11. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-2]. Compute p(-2) exactly.

View answer and explanation-2

Multiply each coefficient by the corresponding natural-number power and add: p(-2)=-2. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0011/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 12. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-1, 1, 1]. Compute p(-1) exactly.

View answer and explanation-1

Multiply each coefficient by the corresponding natural-number power and add: p(-1)=-1. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0012/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 13. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [0, 2, -1]. Compute p(0) exactly.

View answer and explanation0

Multiply each coefficient by the corresponding natural-number power and add: p(0)=0. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0013/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 14. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [1, 3]. Compute p(1) exactly.

View answer and explanation4

Multiply each coefficient by the corresponding natural-number power and add: p(1)=4. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0014/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 15. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [2, -3, 1]. Compute p(2) exactly.

View answer and explanation0

Multiply each coefficient by the corresponding natural-number power and add: p(2)=0. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0015/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 16. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [3, -2, -1]. Compute p(3) exactly.

View answer and explanation-12

Multiply each coefficient by the corresponding natural-number power and add: p(3)=-12. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0016/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 17. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [4, -1]. Compute p(4) exactly.

View answer and explanation0

Multiply each coefficient by the corresponding natural-number power and add: p(4)=0. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0017/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 18. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5, 0, 1]. Compute p(5) exactly.

View answer and explanation30

Multiply each coefficient by the corresponding natural-number power and add: p(5)=30. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0018/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 19. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [6, 1, -1]. Compute p(6) exactly.

View answer and explanation-24

Multiply each coefficient by the corresponding natural-number power and add: p(6)=-24. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0019/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 20. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [7, 2]. Compute p(7) exactly.

View answer and explanation21

Multiply each coefficient by the corresponding natural-number power and add: p(7)=21. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0020/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 21. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [8, 3, 1]. Compute p(8) exactly.

View answer and explanation96

Multiply each coefficient by the corresponding natural-number power and add: p(8)=96. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0021/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 22. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [9, -3, -1]. Compute p(9) exactly.

View answer and explanation-99

Multiply each coefficient by the corresponding natural-number power and add: p(9)=-99. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0022/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 23. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [10, -2]. Compute p(10) exactly.

View answer and explanation-10

Multiply each coefficient by the corresponding natural-number power and add: p(10)=-10. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0023/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 24. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [11, -1, 1]. Compute p(11) exactly.

View answer and explanation121

Multiply each coefficient by the corresponding natural-number power and add: p(11)=121. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0024/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 25. Exact Polynomial Evaluation

In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [12, 0, -1]. Compute p(12) exactly.

View answer and explanation-132

Multiply each coefficient by the corresponding natural-number power and add: p(12)=-132. This is evaluation at one input and does not establish a general polynomial identity.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0025/)

Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate





## 26. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [0], find a_0, the coefficient of x^0.

View answer and explanation0

The index is the nonnegative exponent, so a_0=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0026/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 27. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-9, -1, 2], find a_1, the coefficient of x^1.

View answer and explanation-1

The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0027/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 28. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-8, 0, 3], find a_2, the coefficient of x^2.

View answer and explanation3

The index is the nonnegative exponent, so a_2=3. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0028/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 29. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-7, 1, 4], find a_3, the coefficient of x^3.

View answer and explanation0

The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0029/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 30. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-6, 2, 1], find a_4, the coefficient of x^4.

View answer and explanation0

The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0030/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 31. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-5, -2, 2], find a_0, the coefficient of x^0.

View answer and explanation-5

The index is the nonnegative exponent, so a_0=-5. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0031/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 32. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-4, -1, 3], find a_1, the coefficient of x^1.

View answer and explanation-1

The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0032/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 33. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-3, 0, 4], find a_2, the coefficient of x^2.

View answer and explanation4

The index is the nonnegative exponent, so a_2=4. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0033/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 34. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-2, 1, 1], find a_3, the coefficient of x^3.

View answer and explanation0

The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0034/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 35. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [-1, 2, 2], find a_4, the coefficient of x^4.

View answer and explanation0

The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0035/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 36. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [0, -2, 3], find a_0, the coefficient of x^0.

View answer and explanation0

The index is the nonnegative exponent, so a_0=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0036/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 37. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [1, -1, 4], find a_1, the coefficient of x^1.

View answer and explanation-1

The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0037/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 38. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [2, 0, 1], find a_2, the coefficient of x^2.

View answer and explanation1

The index is the nonnegative exponent, so a_2=1. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0038/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 39. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [3, 1, 2], find a_3, the coefficient of x^3.

View answer and explanation0

The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0039/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 40. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [4, 2, 3], find a_4, the coefficient of x^4.

View answer and explanation0

The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0040/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 41. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [5, -2, 4], find a_0, the coefficient of x^0.

View answer and explanation5

The index is the nonnegative exponent, so a_0=5. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0041/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 42. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [6, -1, 1], find a_1, the coefficient of x^1.

View answer and explanation-1

The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0042/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 43. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [7, 0, 2], find a_2, the coefficient of x^2.

View answer and explanation2

The index is the nonnegative exponent, so a_2=2. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0043/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 44. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [8, 1, 3], find a_3, the coefficient of x^3.

View answer and explanation0

The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0044/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 45. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [9, 2, 4], find a_4, the coefficient of x^4.

View answer and explanation0

The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0045/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 46. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [10, -2, 1], find a_0, the coefficient of x^0.

View answer and explanation10

The index is the nonnegative exponent, so a_0=10. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0046/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 47. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [11, -1, 2], find a_1, the coefficient of x^1.

View answer and explanation-1

The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0047/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 48. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [12, 0, 3], find a_2, the coefficient of x^2.

View answer and explanation3

The index is the nonnegative exponent, so a_2=3. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0048/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 49. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [13, 1, 4], find a_3, the coefficient of x^3.

View answer and explanation0

The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0049/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 50. Coefficient of a Specified Power

In Q[x], coefficient vectors are ordered from lowest to highest power. For [14, 2, 1], find a_4, the coefficient of x^4.

View answer and explanation0

The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0050/)

Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate





## 51. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [0]. Answer undefined for the zero polynomial.

View answer and explanationundefined

All coefficients are zero, so this entry calls the degree undefined. Trailing zeros do not change the polynomial.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0051/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 52. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-4]. Answer undefined for the zero polynomial.

View answer and explanation0

The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0052/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 53. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-10, 0, 1]. Answer undefined for the zero polynomial.

View answer and explanation2

The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0053/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 54. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-9, 0, 0, 1]. Answer undefined for the zero polynomial.

View answer and explanation3

The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0054/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 55. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-8, 0, 0, 0, 1]. Answer undefined for the zero polynomial.

View answer and explanation4

The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0055/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 56. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [2]. Answer undefined for the zero polynomial.

View answer and explanation0

The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0056/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 57. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-6, 2]. Answer undefined for the zero polynomial.

View answer and explanation1

The largest index with a nonzero coefficient is 1, so the degree is 1. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0057/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 58. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-5, 0, 2]. Answer undefined for the zero polynomial.

View answer and explanation2

The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0058/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 59. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-4, 0, 0, 2]. Answer undefined for the zero polynomial.

View answer and explanation3

The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0059/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 60. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-3, 0, 0, 0, 2]. Answer undefined for the zero polynomial.

View answer and explanation4

The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0060/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 61. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [3]. Answer undefined for the zero polynomial.

View answer and explanation0

The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0061/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 62. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [-1, 3]. Answer undefined for the zero polynomial.

View answer and explanation1

The largest index with a nonzero coefficient is 1, so the degree is 1. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0062/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 63. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [0, 0, 3]. Answer undefined for the zero polynomial.

View answer and explanation2

The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0063/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 64. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [1, 0, 0, 3]. Answer undefined for the zero polynomial.

View answer and explanation3

The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0064/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 65. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [2, 0, 0, 0, 3]. Answer undefined for the zero polynomial.

View answer and explanation4

The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0065/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 66. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [4]. Answer undefined for the zero polynomial.

View answer and explanation0

The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0066/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 67. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [4, 4]. Answer undefined for the zero polynomial.

View answer and explanation1

The largest index with a nonzero coefficient is 1, so the degree is 1. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0067/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 68. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [5, 0, 4]. Answer undefined for the zero polynomial.

View answer and explanation2

The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0068/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 69. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [6, 0, 0, 4]. Answer undefined for the zero polynomial.

View answer and explanation3

The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0069/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 70. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [7, 0, 0, 0, 4]. Answer undefined for the zero polynomial.

View answer and explanation4

The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0070/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 71. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [5]. Answer undefined for the zero polynomial.

View answer and explanation0

The largest index with a nonzero coefficient is 0, so the degree is 0. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0071/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 72. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [9, 5]. Answer undefined for the zero polynomial.

View answer and explanation1

The largest index with a nonzero coefficient is 1, so the degree is 1. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0072/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 73. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [10, 0, 5]. Answer undefined for the zero polynomial.

View answer and explanation2

The largest index with a nonzero coefficient is 2, so the degree is 2. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0073/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 74. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [11, 0, 0, 5]. Answer undefined for the zero polynomial.

View answer and explanation3

The largest index with a nonzero coefficient is 3, so the degree is 3. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0074/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 75. Polynomial Degree

In Q[x], coefficient vectors are ordered from lowest to highest power. Find the degree of the polynomial represented by [12, 0, 0, 0, 5]. Answer undefined for the zero polynomial.

View answer and explanation4

The largest index with a nonzero coefficient is 4, so the degree is 4. Trailing zeros do not increase it.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0075/)

Univariate polynomials over the rationals · qf-polynomial-degree · candidate





## 76. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [0] and [0]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0]

Add coefficients of each equal power and remove trailing zeros. The sum is [0]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0076/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 77. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [2, 3] and [-2, -3]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0]

Add coefficients of each equal power and remove trailing zeros. The sum is [0]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0077/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 78. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-10, 3] and [10, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 4]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 4]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0078/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 79. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-9, 4] and [9, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 3]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0079/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 80. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-8, 1] and [8]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 1]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 1]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0080/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 81. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-7, 2] and [7, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 3]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0081/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 82. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-6, 3] and [6, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 2]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0082/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 83. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-5, 4] and [5]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 4]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 4]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0083/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 84. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-4, 1] and [4, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 2]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0084/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 85. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-3, 2] and [3, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 1]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 1]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0085/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 86. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-2, 3] and [2]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 3]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0086/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 87. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [-1, 4] and [1, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 5]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 5]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0087/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 88. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [0, 1] and [0, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0]

Add coefficients of each equal power and remove trailing zeros. The sum is [0]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0088/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 89. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [1, 2] and [-1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 2]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0089/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 90. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [2, 3] and [-2, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 4]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 4]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0090/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 91. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [3, 4] and [-3, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 3]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0091/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 92. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [4, 1] and [-4]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 1]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 1]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0092/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 93. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [5, 2] and [-5, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 3]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0093/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 94. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [6, 3] and [-6, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 2]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0094/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 95. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [7, 4] and [-7]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 4]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 4]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0095/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 96. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [8, 1] and [-8, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 2]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 2]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0096/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 97. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [9, 2] and [-9, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 1]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 1]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0097/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 98. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [10, 3] and [-10]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 3]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 3]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0098/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 99. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [11, 4] and [-11, 1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0, 5]

Add coefficients of each equal power and remove trailing zeros. The sum is [0, 5]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0099/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate





## 100. Coefficientwise Polynomial Addition

In Q[x], coefficient vectors are ordered from lowest to highest power. Add [12, 1] and [-12, -1]. Return the coefficient vector with trailing zeros removed, representing the zero polynomial by [0].

View answer and explanation[0]

Add coefficients of each equal power and remove trailing zeros. The sum is [0]; the zero polynomial is represented by [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-r02-polynomial-0100/)

Univariate polynomials over the rationals · qf-polynomial-addition · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-r02-polynomial-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261003-r02-polynomial-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-r02-polynomial-foundations/validation-report.json)