Let $p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$ with $a_0,a_1,\dots,a_n \in \Bbb R$. Prove that if $a_0a_n < 0$, then $p$ has a positive root.
2408
Show that a polynomial has at least one positive solution/root.
19046
Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
2907
How do I show that $x$ is a root of $p(x)$ in $F[x]/\langle p(x)\rangle$?
118039
How prove $p!|(a+1)$ if $p!|a^p+1$
55210
Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1.
154507
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
114872
If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$
72410
If $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the roots of the equation $x^n+1$
88662
$n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.
158844
If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
82591
Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
156034
sufficient condition for a polynomial to have roots in $[0,1]$
120954
Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
122115
Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
81778
If $m|n$ and $a$ is a primitive root of $n$, show that $a$ is a primitive root of $m$ (understanding a tip)
16660
Prove that any positive number has an $n$th root
133228
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.