query: Prove that there exists a polynomial p(x) with coefficients belonging to the set {-1, 0, 1} such that p(3) = n, for some positive integer n
positive_passages.docid: 48308
positive_passages.text: Uniqueness of radix representation of integers in base 3
negative_passages.docid: 148451
negative_passages.text: Polynomial inequality proof
negative_passages.docid: 7199
negative_passages.text: Suppose that $P(x)$ is a polynomial of degree $n$ such that $P(k)=\dfrac{k}{k+1}$ for $k=0,1,\ldots,n$. Find the value of $P(n+1)$
negative_passages.docid: 112205
negative_passages.text: Which of the following polynomials are subspaces of $\mathbb{P}_n$ for an appropriate value of n?
negative_passages.docid: 148217
negative_passages.text: A polynomial with integer coefficients that attains the value $5$ at four distinct points
negative_passages.docid: 129834
negative_passages.text: Show that there is a unique polynomial of degree at most $2n+1$ such that $q^{[k]}(x_1)=a_k,$ $q^{[k]}(x_2)=b_k$ for $k=0, \dots, n$.
negative_passages.docid: 99835
negative_passages.text: Polynomial in $\mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$
negative_passages.docid: 75349
negative_passages.text: Minimal polynomial of $\sqrt[3]{2} + \sqrt{3}$
negative_passages.docid: 59535
negative_passages.text: Prove that there exists such a set of $n$ positive integers
negative_passages.docid: 67291
negative_passages.text: Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
negative_passages.docid: 127547
negative_passages.text: Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$.
negative_passages.docid: 33011
negative_passages.text: Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
negative_passages.docid: 129109
negative_passages.text: Is the set of polynomial with coefficients on $\mathbb{Q}$ enumerable?
negative_passages.docid: 85287
negative_passages.text: Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$
negative_passages.docid: 53244
negative_passages.text: Prove that $p \in \mathbb{R}[x]$ can be represented as a sum of squares of polynomials from $\mathbb{R}[x]$
negative_passages.docid: 12079
negative_passages.text: A positive polynomial is the sum of two squares in $\mathbb{R}[X]$
negative_passages.docid: 116856
negative_passages.text: Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$