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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.
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [0]. Compute p(-12) exactly.
0
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [5]. Compute p(-11) exactly.
5
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [0, 1]. Compute p(-10) exactly.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-8, 1]. Compute p(-8) exactly.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
28
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-5, -3]. Compute p(-5) exactly.
10
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
20
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [-2]. Compute p(-2) exactly.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
0
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [1, 3]. Compute p(1) exactly.
4
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
0
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [4, -1]. Compute p(4) exactly.
0
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
30
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [7, 2]. Compute p(7) exactly.
21
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
96
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. The integer coefficients of p are [10, -2]. Compute p(10) exactly.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
121
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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
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.
-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
Univariate polynomials over the rationals · qf-polynomial-exact-evaluation · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [0], find a_0, the coefficient of x^0.
0
The index is the nonnegative exponent, so a_0=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-9, -1, 2], find a_1, the coefficient of x^1.
-1
The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-8, 0, 3], find a_2, the coefficient of x^2.
3
The index is the nonnegative exponent, so a_2=3. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-7, 1, 4], find a_3, the coefficient of x^3.
0
The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-6, 2, 1], find a_4, the coefficient of x^4.
0
The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-5, -2, 2], find a_0, the coefficient of x^0.
-5
The index is the nonnegative exponent, so a_0=-5. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-4, -1, 3], find a_1, the coefficient of x^1.
-1
The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-3, 0, 4], find a_2, the coefficient of x^2.
4
The index is the nonnegative exponent, so a_2=4. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-2, 1, 1], find a_3, the coefficient of x^3.
0
The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [-1, 2, 2], find a_4, the coefficient of x^4.
0
The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [0, -2, 3], find a_0, the coefficient of x^0.
0
The index is the nonnegative exponent, so a_0=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [1, -1, 4], find a_1, the coefficient of x^1.
-1
The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [2, 0, 1], find a_2, the coefficient of x^2.
1
The index is the nonnegative exponent, so a_2=1. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [3, 1, 2], find a_3, the coefficient of x^3.
0
The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [4, 2, 3], find a_4, the coefficient of x^4.
0
The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [5, -2, 4], find a_0, the coefficient of x^0.
5
The index is the nonnegative exponent, so a_0=5. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [6, -1, 1], find a_1, the coefficient of x^1.
-1
The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [7, 0, 2], find a_2, the coefficient of x^2.
2
The index is the nonnegative exponent, so a_2=2. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [8, 1, 3], find a_3, the coefficient of x^3.
0
The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [9, 2, 4], find a_4, the coefficient of x^4.
0
The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [10, -2, 1], find a_0, the coefficient of x^0.
10
The index is the nonnegative exponent, so a_0=10. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [11, -1, 2], find a_1, the coefficient of x^1.
-1
The index is the nonnegative exponent, so a_1=-1. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [12, 0, 3], find a_2, the coefficient of x^2.
3
The index is the nonnegative exponent, so a_2=3. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [13, 1, 4], find a_3, the coefficient of x^3.
0
The index is the nonnegative exponent, so a_3=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
In Q[x], coefficient vectors are ordered from lowest to highest power. For [14, 2, 1], find a_4, the coefficient of x^4.
0
The index is the nonnegative exponent, so a_4=0. Coefficients beyond the finite support are zero.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-coefficient · candidate
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.
undefined
All coefficients are zero, so this entry calls the degree undefined. Trailing zeros do not change the polynomial.
Reviewed step-by-step derivation
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
0
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
2
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
3
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
4
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
0
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
1
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
2
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
3
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
4
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
0
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
1
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
2
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
3
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
4
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
0
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
1
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
2
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
3
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
4
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
0
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
1
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
2
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
3
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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.
4
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
Univariate polynomials over the rationals · qf-polynomial-degree · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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].
[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
Univariate polynomials over the rationals · qf-polynomial-addition · candidate
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