query: Prove that there exists at least one root of $g$ between any two roots of $f$
positive_passages.docid: 146572
positive_passages.text: Prove that between any two roots of $f$ there exists at least one root of $g$
negative_passages.docid: 13270
negative_passages.text: Prove $\exists \xi \in (a,b):f'(\xi)<1+f^2(\xi)$
negative_passages.docid: 115299
negative_passages.text: Elements of $\mathbb{F}_p$ having cube roots in $\mathbb{F}_p$
negative_passages.docid: 10859
negative_passages.text: Prove $\sup(f+g) \le \sup f + \sup g$
negative_passages.docid: 2408
negative_passages.text: Show that a polynomial has at least one positive solution/root.
negative_passages.docid: 31522
negative_passages.text: Proving $f$ has at least one zero inside unit disk
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
negative_passages.docid: 107765
negative_passages.text: Prove that there exists two roots in $[-1,1]$
negative_passages.docid: 143321
negative_passages.text: Number of roots of a polynomial over a finite field
negative_passages.docid: 142000
negative_passages.text: Prove that there is an irrational number and a rational number between any two distinct real numbers
negative_passages.docid: 82921
negative_passages.text: Uniqueness of $k$th root mod $m$ if $(k, \phi(m)) = 1$.
negative_passages.docid: 148585
negative_passages.text: Show that, for any $\epsilon>0$, there exist two rationals such that $q < x < q'$ and $|q-q'|<\epsilon$
negative_passages.docid: 40609
negative_passages.text: Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
negative_passages.docid: 156034
negative_passages.text: sufficient condition for a polynomial to have roots in $[0,1]$
negative_passages.docid: 146312
negative_passages.text: Is the set of roots with respect to their values and multiplicies unique for each $g(x)$ in $a_n x^n + g(x)$?
negative_passages.docid: 23712
negative_passages.text: $F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.
negative_passages.docid: 13656
negative_passages.text: Prove existence of a real root.