query: If $ax^2+(b+c)x+d+e=0$ has roots greater than $1$, then show that $ax^4+bx^3+cx^2+dx+e$ has at least one real root
positive_passages.docid: 18702
positive_passages.text: If $ax^2+(c+b)x+e+d=0$ has real roots greater than $1$, show that $ax^4+bx^3+cx^2+dx+e$ has at least one real root
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 55582
negative_passages.text: How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
negative_passages.docid: 24000
negative_passages.text: Roots of Sum of Two Polynomials (with Known Roots)
negative_passages.docid: 141184
negative_passages.text: Proof that a degree 4 polynomial has at least two roots
negative_passages.docid: 39779
negative_passages.text: If $a,b,c>0$ and $a+b+c=1$, prove that $\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ac}{b+ac}} \le \frac{3}{2}$.
negative_passages.docid: 86378
negative_passages.text: Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$
negative_passages.docid: 154507
negative_passages.text: If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts
negative_passages.docid: 63409
negative_passages.text: If $x^2+ax+b=0$ has a rational root, show that it is in fact an integer
negative_passages.docid: 135385
negative_passages.text: $0 < a_0 \leq a_1 \leq \cdots \leq a_n$, show $ p_n(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n = 0$ has no root in $|z| < 1$
negative_passages.docid: 112334
negative_passages.text: Real roots of a polynomial of real co-efficients , with the co-efficients of $x^2 , x$ and the constant term all $1$
negative_passages.docid: 55210
negative_passages.text: Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1.
negative_passages.docid: 15730
negative_passages.text: Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution
negative_passages.docid: 63689
negative_passages.text: proving roots of equation $x^2+x+4f=0$ are real.
negative_passages.docid: 126445
negative_passages.text: Proving the second root of a quadratic equation
negative_passages.docid: 100571
negative_passages.text: Prove $(1-a)(1-b)(1-c)(1-d)\geq abcd$ if $a^2+b^2+c^2+d^2=1$