# Tagged Questions

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### If a polynomial has only real zeros then $a_{0}+a_{1}+\cdots+a_{n}\le\frac{(n+1)^n}{\binom{n}{s}(n-s)^{n-s}(s+1)^s}\cdot\max_{k}a_{k}$

Question: For all real polynomials $P(x)=a_{0}+a_{1}x+\cdots+a_{n}x^n$ of degree $n$, with only real zeros,we have ...
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### How to solve the inequality: $\prod_{k=1}^N\left(x^k-k^2\right)\gt0$

Given the inequality: $$\displaystyle\prod_{k=1}^N\left(x^k-k^2\right)\gt0$$ how can I solve it? I suppose there is a difference if $N=2n$ or $N=2n+1$ with $n\in\mathbb{N}$, but I'm unable to find a ...
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### prove $\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$

Show that if $a,b,c,d \geq 0$ and $ab+bc+cd+da=1$ :$$\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$$ yet again it should be solved with Cauchy inequality. thing i have done so far: ...
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### If $a^2=b^2+c^2$ and $0<n<2$ prove $a^n<b^n+c^n$

If $a^2=b^2+c^2$ and $a,b,c$ are positive real numbers, prove (a) if $n>2$ then $a^n>b^n+c^n$, (b) if $0<n<2$ then $a^n<b^n+c^n$. Part (a) was easy to prove: $a^2=b^2+c^2$ and ...
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### How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
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### solving the inequalty

are there any ways to solve :$x^4 -6x^3 +28x^2 -64x +96 >0$ ?
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### Polynomial factors involving inequalities

How to factorise the polynomial $p(x) = x^4-2x^3 + 2x - 1$. Hence, solve the inequality $p(x) \gt 0$ ?
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### Finding matrix index from triangular array offset

I have a mapping from a lower triangular matrix, A, to a vector,v: A(i,j) -> v( $\lfloor i(i+1)/2 \rfloor + j$ ) $i,j\in[0,N]$, $j\leq i$, $N\in\cal{N}$, $N\geq 0$ (so, my first row is row 0, and ...
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### $­\prod_{k=1}^{n}(x_k-a)(x_k-b)\leqslant\sum_{q=1}^{n}x_q^2\prod_{p=1,p\neq q}^{n}(x_p-a)(x_p-b).$?

Is there a name for this formula? For $f_k,w_k\geqslant0$. $$­\prod_{k=1}^{n}f_k\leqslant\sum_{q=1}^{n}w_q\prod_{p=1,p\neq q}^{n}f_p.$$ I believe that there is $w_k$ that make the formula true. Am I ...
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### Cauchy-Schwarz Inequality [closed]

Hi,I want to prove above unequal.All elements are positive or bigger than zero. The way to prove it is not important, but because I weak in mathematical i want to know all thing about details.
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### How prove this $p(x)>0$ if $p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$

let the polynomials $$p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$$ and such $$a_{0}+\sum_{a_{i}<0}(1-\dfrac{i}{n})\binom{n}{i}a_{i}>0$$ and ...
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### Inequality for quartic polynomial depending on a parameter

Let $f(x) = \frac 14 x^4 - \frac \alpha2 x^2 - (\alpha-1) x - \frac \alpha 2 + \frac 3 4$. I want to show that there exists an $\alpha>1$ such that $f(x)\geq 0$ for $x\leq 0$. Even more, it ...
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### What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
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### Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
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### Determining equality of sets defined by polynomial inequalities in several variables

Let's say I have two sets $S_1,S_2 \subseteq \mathbb{R}^n$ each defined by a number of polynomial inequalities. Is there a computationally feasible way to find whether $S_1 = S_2$? In particular, is ...
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### Finding the minimum value of a 6th degree polynomial algebraically

Is it possible to answer this question using methods of basic algebra? Find the least value of the expression $a^6 + a^4 - a^3 - a + 1$ for real value of $a$. This question is from the 2013 Philippine ...
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### How to simplifying and solving this polynomial?

I have a problem with simplifying the polynomial. In the first time, I see that this polynomial is quite simple, but when I'm trying, I realized that this polynomial isn't as easy as I saw. Here is ...
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### Inequalities with cubic polynomials

So I was checking my friend's notes and found this: Find the interval for which $x^3-x^2-x+1>0$ is true. We haven't been taught to factor cubic polynomials (the rest of the exercises are with ...
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### Inferring a characteristic of a ratio of functions from the ratio of their derivatives

This is a strange one, but I need help trying to understand whether there is any logic behind this or not. Given $\frac {f(\sqrt{2})}{g(\sqrt{2})}=2$, and $\frac {f'(x)}{g'(x)}>2$ for all ...
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### Polynomial inequality proof

Prove $(1-a)(1-b)(1-c)(1-d)>1-a-b-c-d$ and $a,b,c,d$ are real and between 0 and 1. I can do this with $$(1-a)(1-b)>1-a-b \\ 1-a-b+ab>1-a-b \\ ab>0$$ But with $c$ and $d$, this ...
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### A polynomial has only real roots and all coefficients $\pm 1$. Prove the degree $<4$.

Let $P(x)$ be a polynomial with only real roots and all coefficients equal to $\pm 1$. Prove that the degree of the polynomial is less than 4. This is practice for Putnam, but I am not certain where ...
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### verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
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### looking for help proving conjecture about the sign of a polynomial, or finding counterexample

Math people: I would like some help proving this conjecture, which is backed up by a lot of experiments, or finding a counterexample. Let $n \geq 1$ be an integer, $a_0, \ldots, a_n$ be real ...
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### Annoying Polynomial Inequality

Suppose we have a polynomial satisfying $p+p''' \geq p'+p''$ for all $x$. Then $p(x)\geq 0$ for all $x$. I've been stuck on this problem for weeks. The best I can do is supposing there exists $x$ so ...
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### $x_1+x_2+\cdots+x_n\leq M$: Cardinality of Solution Set is $C(M+n, n)$

Show that the number of solutions in nonnegative integers of the inequality $$x_1+x_2+\cdots+x_n\leq M,$$ where $M$ is a nonnegative integer, is $C(M+n, n)$.
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### How prove this polynomials inequality

Let $f=a_0x^n+a_1x^{n-1}+\ldots+a_{n-1}x+a_n\in \mathbb{R}[X]$ a polynomial which has the roots contained in $(-1,1)$. Prove that: $$\left|\frac{a_1+a_3+a_5+\ldots}{a_0+a_2+a_4+\ldots}\right|<1$$ ...
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### Prove that $x^{12}-x^9+x^4-x+1>0$ for all $x\in \mathbb{R}$

Prove that the expression $x^{12}-x^9+x^4-x+1>0\; \forall x\in \mathbb{R}$ My try:: Using Interval method:: $\bullet \;$If $x\leq 0$, Then $x^{12}-x^9+x^4-x+1>0$ $\bullet \;$If ...
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### how to find bounds on (complex) coefficients from bounds on a polynomial?

I'm trying to prove the following two statements about a polynomial $p$ of degree $n$ with complex coefficients: If $|p(x)|\le1$ for all real $x$ with $|x|\le1$, then every coefficient of $p$ has ...
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### Lower bound for polynomial with complex coefficient

Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{1}z+a_{0}$ be a polynomial with complex coefficients. Define $R:=1+\sum_{k=0}^{n-1}|a_k|$. Show that $|p(z)| > R$ for all $z \in \mathbb C$ with $|z|>R$. ...
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### Polynomial always non-negative

Is there an elegant way to demonstrate that (for example) $x^{2016}-1008x^2+1007\ge 0$ $\forall x\in \mathbb{R}$ ? I tried to write it as sum of squares, but I didn't succeed.
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### Half-symmetric, homogeneous inequality

Let $x,y,z$ be three positive numbers. Can anybode prove the follwing inequality : $(x^2y^2+z^4)^3 \leq (x^3+y^3+z^3)^4$ (or find a counterexample, or find a reference ...)
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### How to solve systems of polynomial inequalities?

I am currently working on a project that deals with systems of inequalities and so far I have found algorithms for the basic case of a system of inequalities as well as the non-strict linear ...
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### Polynomial bound

Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that $$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$ Suppose that $P(x)> 0$ for all ...
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### Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that:

Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that: $|a_{n}|+|a_{n-1} | \leqslant 2^{n-1}$. ...
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### Minimum value of a polynomial of degree 2.

Let $f:\mathbb R\to \mathbb R$ be a map defined as $f(x)=ax^2+bx+c$, we can rewrite this map as $f(x)=a(x+\frac{b}{2a})^2-\frac{\Delta}{4a}$, where $\Delta=b^2-4ac$. If $a\gt 0$, then the minimum ...
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Let $a,b$ be two positive numbers such that $a^3 \gt 27b$. Consider the polynomial $$W(x)=x^3-2ax^2+a^2x-4b$$ Then we have $$W(0)=-4b \lt 0, \ W(\frac{a}{3})=\frac{4}{27}(a^3-27b) \gt 0, \ ... 2answers 76 views ### \sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1 Given that the equation$$p(x)=a_0x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$has n distinct positive roots, prove that$$\sum_{i=1}^{n-1} \left|\dfrac{a_ia_{n-i}}{a_n}\right| \geq C_{2n}^n-1 I had ...
In my course I have come across the following problem: The Chebyshev polynomial of degree $n$, $T_n(x)$, is defined on $[-1,1]$ by $T_n(x)=\cos n\theta$. Let $q_{n+1}(x)$ be any monic ...
In a proof, I arrive at three inequalities for all $p,q \geqslant 0$: \begin{align} \frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 + \frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr ...