query: Let $p(x) = a_0 + a_1 x + .. + a_n x^{n}$. Assume $a_n \neq 0$ and n is odd. Prove $\exists$ x such that p(x)=0?
positive_passages.docid: 13981
positive_passages.text: Proof that every polynomial of odd degree has one real root
negative_passages.docid: 1487
negative_passages.text: How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
negative_passages.docid: 111805
negative_passages.text: If $a_{1}=1$ and $a_{n+1}=1+\frac{n}{a_{n}}$, then $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)$
negative_passages.docid: 13083
negative_passages.text: If for a polynomial $P(k) = 2^k$ for $k = 0, 1, . . . , n$, what is $P(n+1)$?
negative_passages.docid: 157847
negative_passages.text: If $x_n \to 0$, show that $\sqrt{x_n} \to 0$
negative_passages.docid: 125707
negative_passages.text: A simple proof (hopefully) for $2^x \neq 0$
negative_passages.docid: 21941
negative_passages.text: Proving: $x = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$
negative_passages.docid: 143481
negative_passages.text: If $P(n)$ divides $P(P(n)-2015)$, prove that $P(-2015)=0$
negative_passages.docid: 134348
negative_passages.text: When is $(p - 1)! + 1$ a power of $p$?
negative_passages.docid: 144165
negative_passages.text: $n\in N, a_{n+1} = a_n (1 - a_n)$ , $0 < a_0 < 1$. Prove that $\lim\limits_{n\to\infty} a_n\cdot n = 1$.
negative_passages.docid: 62239
negative_passages.text: For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
negative_passages.docid: 39843
negative_passages.text: Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$
negative_passages.docid: 42300
negative_passages.text: Let the sequence be defined recursively $X_{n+1}=pX_{n}+q$ for p nonzero and $X_{1}$ arbitrary. In what conditions does the sequence converge?
negative_passages.docid: 85977
negative_passages.text: Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 97656
negative_passages.text: Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$
negative_passages.docid: 129331
negative_passages.text: Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$