# 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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### Prove that $f(x) = x^4+ax^3+bx^2+cx+d$ does not has all rational roots

The quartic polynomial $f(x) = x^4 + a x^3 + b x^2 + c x + d$ is such that $ad$ is odd and $bc$ is even. Prove that $f(x)$ does not has all rational roots. My attempt: Clearly, f(x) will have ...
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### root of $g$ is smaller than that of $f$.

For a fixed natural no. $n\ge4$, consider $$f(x)=x^3-(n+2)x^2+2nx-2,$$ $$g(x)=x^3-(n+3)x^2+2(n+1)x-2,$$ It seems that smallest root of $g$ is smaller than that of $f$. Can someone show how to prove it....
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### Cubic polynomial with 1 real root and 2 complex conjugated roots (real coefficients)

I am stuck on this problem about cubic polynomials. I rely on the Wikipedia page on the topic. Using wikipedia notations (chapter "General formula for roots") : For the case where $\Delta > 0$, ...
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### Proving that largest root (obtained via P.C.A.) is a symmetric function

Suppose, we are given $\textbf{X} = (X_1, X_2, \ldots,X_m)$ and $\textbf{Y} = (Y_1, Y_2, \ldots, Y_n)$. Also, we are given, S = pooled variance. If we implement Principal Component Analysis (P.C.A.) ...
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### Find the condition such that the roots of the polynomial are in AP

$f(x)=x^3+3px^2+3qx+r$ has roots in AP.Find the relation between $p,q$ and $r$. [Answer:$-2p^2-3pq+r=0$] My attempt:- Taking $d$ as the common difference of the roots in AP we ...
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### Graphically solving for complex roots — how to visualize?

So recently we've been doing the complex roots of quadratics, cubics and polynomials in general in school. But my question is, is there a way to see where these roots are, just like you can see where ...
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### Explicit form of a generating function.

Let $q \geq p$ be natural numbers both larger than or equal to two. Let $u(z):=z^p+z^{p+1}+...+z^q$ and $p(z)=\frac{z u'(z)}{1-u(z)}$. Since $p(z)$ is rational, one can write (by the theory of ...
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### Iteration of polynomial has only positive roots

Let $P(x)$ be a real polynomial with a positive leading coefficient, and $k\geq 2$ an integer. Suppose that $Q(x)=P(P(\dots(P(x))\dots))$, where there are $k$ iterations of $P$'s, has at least one ...
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### Solve $z^6+7z^3-8=0$

I want to find the solutions $z^6+7z^3-8=0$ but I don't know where to start because of the high degree of the equation. This is an exercise that involves complex numbers, so I have to transform the ...
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### Roots of $x^{101}-100x^{100}+100=0$

I do not know how to prove that $x^{101}-100x^{100}+100=0$ has exactly two positive roots. Some can give me hint for solving this please. Thanks for your time.
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Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ... 0answers 55 views ### Find a polynomial such that this proposed root finding algorithm fails. Is this polynomial root finding algorithm below known, and under what conditions for the choice of polynomial coefficients does it find at least one root? Description of the algorithm: Consider the ... 1answer 54 views ### What mathematical notation can use for this formula I just played around with archimedes \pi formula and ended up with \pi = \lim\limits_{n \to \infty} 6 \cdot 2^n \cdot \sqrt{2 - \sqrt{2 + \sqrt{2 + ...n times... \sqrt{2 + \sqrt{3}}}}} I want to ... 0answers 34 views ### Is there an analytic solution to find zeroes of a polynomial plus sin()? Is there an analytic solution to find the zeroes of an equation of the form:$$0 = at^2+bt+c+\sin(mt^2+nt+o)$$1answer 32 views ### When are the limits of roots of a polynomial identical to the roots of the limit of the polynomial? I have a univariate polynomial of degree n (where n is larger than 4). The real-valued coefficients of the polynomial depend on a parameter \psi, i.e.$$p_\psi(x)=a_n(\psi) x^n+a_{n-1}(\psi) x^...
Let $a>0$ be a fixed parameter. I would like to find the (I think there are only two) $x\in \mathbb{R}$ such that $$(x-a)e^{-\frac{1}{2}(x-a)^2} = (x+a)e^{-\frac{1}{2}(x+a)^2}.$$ I know this might ...