query: Factor $x^6+x^5+x^4+x^3+x^2+x+1$ in $\mathbb{F}_2[x]$
positive_passages.docid: 157031
positive_passages.text: Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$
negative_passages.docid: 29963
negative_passages.text: $6$ divides $a+b+c$ if and only if $6$ divides $a^{3}+b^{3}+c^{3}$
negative_passages.docid: 130027
negative_passages.text: Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$
negative_passages.docid: 46292
negative_passages.text: Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$
negative_passages.docid: 153111
negative_passages.text: factoring a quadratic equation
negative_passages.docid: 9217
negative_passages.text: Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$.
negative_passages.docid: 71572
negative_passages.text: factor-square property (FSP) of polynomials
negative_passages.docid: 129946
negative_passages.text: Synthetic division of polynomials by factor of the form $(ax+b)$
negative_passages.docid: 85459
negative_passages.text: $x^3+x+1$ irreducible over $\mathbb{F}_2$, $\theta$ root, compute powers of $\theta$
negative_passages.docid: 96875
negative_passages.text: Finding the value of $x^{x^2}+x^{x^6}$ given that $x^{x^4}=4$
negative_passages.docid: 97652
negative_passages.text: Can every real polynomial be factored up to quadratic factors?
negative_passages.docid: 107133
negative_passages.text: Let f(x) be a polynomial satisfying $\lim_{x \to\infty} \frac{x^4f(x)}{x^8+1}=3,$ $f(2)=5 $ and $f(3)=10$, $f(-1)=2$ and $f(-6)=37$
negative_passages.docid: 81669
negative_passages.text: The logic of the AC Method of factoring quadratic polynomials
negative_passages.docid: 85525
negative_passages.text: Prove that $x^3+x+1$ splits in $Z_2[x]/(x^3+x+1)$
negative_passages.docid: 146887
negative_passages.text: Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible
negative_passages.docid: 130825
negative_passages.text: How to factor intricate polynomial $ ab^3 - a^3b + a^3c -ac^3 +bc^3 - b^3c $
negative_passages.docid: 121404
negative_passages.text: How to factor polynomials in $\mathbb{Z}_n$?