# Tagged Questions

Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (radicals) and (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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### Roots of polynomial with positive coefficients

My question is very simple. Suppose we have a polynomial defined as follows: $$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots+a_0$$ where all of the $a_n$'s are all real and positive. Is there something ...
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### What is the general solution of a multivariate quadratic equation

There exists a general solution to solve the general quadratic equation $$ax^2 + bx + c = 0$$ Solution: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Does there exist a general solution for a general ...
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### Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + … + 1/(x+a_n) = 1/x$ are all real

Prove that the roots of the equation: $$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$ are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.
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### Evaluate the expression $\sqrt{6-2\sqrt5} + \sqrt{6+2\sqrt5}$

$$\sqrt{6-2\sqrt5} + \sqrt{6+2\sqrt5}$$ Can anyone tell me the formula to this expression. I tried to solve in by adding the two expression together and get $\sqrt{12}$ but as I insert each ...
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### Zeros of a polynomial. [on hold]

If $F(i, x)$ is a polynomial where $i$ is a parameter and $\rho$ is the largest root of $F(0,x)$ and $F(i+1,x)\ge F(i, x)$, Prove that as $i$ increases $\rho$ will increase. I don't understand ...
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### Find $m$ so that given equation has real roots

We are given the following equation: $$x^4 - (2m - 1)x^2 + 4m - 5 = 0, m \in \mathbb{R}$$ Find all $m$'s so that the given equation has real roots. I thought I only had to put $\Delta \geq 0$. ...
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### Find the number of solutions of $x^2+1=2^x$

I tried solving this equation $$x^2+1=2^x$$ but I was able to get only two roots , i.e. $x=0,1$ but the answer given said the equation has 3 roots when I looked at the graph given in the solution it ...
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### Express $c$ and $d$ in terms of $m$ where $c$ and $d$ are zeroes of $f$ where $m > -2$

Let $$f(x) = x^2 - mx -(6m^2+25m+25)$$ where $m > - 2$ It can be shown that $f(x)$ has two zeroes. Suppose we have $c,d \in \mathbb R$ s.t. $c < d$ and $f(c) = f(d) = 0$, express $c$ and $d$ ...
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### Are all the zeros of $1-a_2x^2+a_4x^4-a_6x^6+\cdots$ real for $a_{2n}>a_{2(n+1)}$ with $a_{2n+1}=0$ and $a_{2n}>0$?

This question is related to a previous question of mine. I was not pleased about the conditions I provided there. I had something different in mind but I failed in stating it. So here are the ...
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### Solve Equation with max integer [closed]

Solve please $\dfrac{\left[\sqrt{x-[x ]}\right]}{(x+3)(x+4)}\ \geq0$ edit
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### Is this equation $(n+1)~x^{2n+1}-n~x^{2n}-n=0$ solvable in radicals for some $n \geq 2$?

Consider this polynomial equation: $$(n+1)~x^{2n+1}-n~x^{2n}-n=0,~~~~n \geq 2,~~~n \in \mathbb{N}$$ It's related to another question of mine, but I don't think the context matters here. I'm ...
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### Can a polynomial of $n$ degree have $n+1$ distinct real roots?

Question : Let $f(x) = \sum^n_{k=0}c_kx^k$ be a polynomial function then prove that if $f(x) = 0$ for $n+1$ distinct real values, then every coefficient $c_k$ in $f(x)$ is $0$ , thus $f(x) = 0$ for ...
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### Connecting First Passage Time to Power Spectrum

Let $f$ be a real function. Is there a connection between The first positive abscissa for which its autocorrelation function is equal to zero (which I call the first passage time, fpt) The largest ...
### If $w$ is an imaginary cube root of unity, then the polynomial whose roots are $2w+3w^2$ and $2w^2 + 3w$ is?
What polynomial with complex coefficients has the following as its roots? $2w+3w^2$ and $2w^2 + 3w$ I have tried doing this all the ways I know of, still can't get my pen over it... Can you guys ...