query: Prove that the following polynomial has at least one zero in the interval $(0,1)$ by using Rolle's theorem
positive_passages.docid: 25981
positive_passages.text: Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$
negative_passages.docid: 14836
negative_passages.text: A polynomial sequence
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 45811
negative_passages.text: How to prove that r is a double root if and only if it is a root of a polynomial and of its derivative.
negative_passages.docid: 118873
negative_passages.text: Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
negative_passages.docid: 103246
negative_passages.text: Prove that a polynomial of degree $d$ has at most $d$ roots (without induction)
negative_passages.docid: 124380
negative_passages.text: existence of sequence of polynomial
negative_passages.docid: 6484
negative_passages.text: Polynomials in one variable with infinitely many roots.
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: 128843
negative_passages.text: Rolle's Theorem Contradiction
negative_passages.docid: 141184
negative_passages.text: Proof that a degree 4 polynomial has at least two roots
negative_passages.docid: 33696
negative_passages.text: A polynomial is zero if it zero on infinite subsets
negative_passages.docid: 109102
negative_passages.text: Determine all real polynomials with $P(0)=0$ and $P(x^2+1)=(P(x))^2+1$
negative_passages.docid: 89354
negative_passages.text: Polynomials, prove exercise question about question
negative_passages.docid: 122076
negative_passages.text: A one line proof that one is zero using Mathematica 10
negative_passages.docid: 121771
negative_passages.text: Proof $f(x)\equiv 0$
negative_passages.docid: 127547
negative_passages.text: Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$.