# 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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### How do you solve 5th degree polynomials?

I looked on Wikipedia for a formula for roots of a 5th degree polynomial, but it said that by Abel's theorem it isn't possible. The Abel's theorem states that you can't solve specific polynomials of ...
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### Showing that an equation has no solution in $\Bbb Z$

Show that $x^3 + 10x^2 + 6x + 2 = 0$ has no solutions in $\Bbb Z$. This seems rather trivial to do but I don't know how to rigorously show this is true. Having graphed this and attempted to factor,I ...
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### Constructible Solutions

We know that if a cubic equation with a rational coefficients has a constructible root, then the equation has a rational root. Now let; $$x^3-2x+2\sqrt{2}=0$$ Could $\sqrt{2}t$ be a viable ...
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### What are the roots of $x^{2} = 2^{x}$? [duplicate]

What are the roots of $x^{2} = 2^{x}$? I drew the graphs and found $x = 2$ and $x = 4$, and there is one other root in $[-1,0]$. Can anyone describe an algebraic method to obtain all roots?
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### Hermite-Bielher Theorem for entire functions

In a question asked by Bobby Ocean, the following theorem is cited: Hermite-Kakeya Theorem(for polynomials) - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros ...
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### Nonzero root of equation

Is there a condition so that a polynomial has at least one nonzero root? Suppose we have the equation $$\sum_{i=0}^m \alpha_i x^i=0$$ Do the coefficients have to satisfy a specific condition so ...
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### On discriminants and nature of an equation's roots?

Edited: All equations in the post are assumed to have all real coefficients and are minimal polynomials. While trying to ascertain if the Brioschi quintic $B(x)=x^5-10cx^3+45c^2x-c^2=0$ could ever ...
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### Factoring the expression $(\sqrt{x^2} -a)^2 + M = 0$

Where, M stands for all other terms in the equation. This is a typical format that you'll see when taking affine sections of an n-torus. I think I figured out how to do it correctly, without violating ...
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### Given that the roots of the equation $9x^2+bx+4=0$ are $4a$ and $a$ and that $b>0$, find the value of $b$.

Given that the roots of the equation $9x^2+bx+4=0$ are $4a$ and $a$ and that $b>0$, find the value of $b$.
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### $f$ has zero of multiplicity $m$ at $\alpha$ and $k$ at $\beta$, where $m+k-1=n$. Prove that $f^{(n)}$ has at least one zero in $(\alpha, \beta)$.

Let $f\in C^n[a,b]$. Suppose that $f$ has a zero of multiplicity $m$ at $\alpha$ and a root of multiplicity $k$ at $\beta$, where $m\geq1$, $k\geq1$, and $m+k-1=n$. Prove that $f^{(n)}$ has at least ...
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### What is the number of zeros of antiderivatives of $(x-1)(x-2)^2(x-3)^3(x-4)^4$?

For each $x \in \mathbb{R}$, let $f(x) = (x-1)(x-2)^2(x-3)^3(x-4)^4$. This defines a function $f : \mathbb{R} \to \mathbb{R}$. There is a unique natural number $k$ such that every antiderivative of ...
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