# Questions tagged [roots]

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 the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

4,070 questions
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### New roots from old

The roots of $2x^2 − 8x − 1$ are $\alpha$ and $\beta$. Suppose another quadratic, $x^2 + qx + r$, has roots $1/(\alpha^3\beta)$ and $1/(\beta^3\alpha)$. What are $q$ and $r$? What I did to solve this ...
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### Number of roots of $p_n-\lambda p_{n-1}$ where $p_n$ are orthogonal polynomials

Let $(p_n)_{n\in\Bbb N}$ be the orthonormal sequence of polynomials associated to a tempered weight $w$ on an interval $I$ (so that for example, $\deg(p_n)=n$). Show that $p_n-\lambda p_{n-1}$ has $n$ ...
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### Methods of solving roots involving matrix determinant

I have a square matrix $F$ whose elements depend non-linearly on a complex parameter $s$. I would like to know the values of $s$ such that $\det(F)=0$, i.e., those $s$ that make $F$ singular. I would ...
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### Prove that the roots of $0=z^3+3z+5$ are outside the circle $S(0,1)$

Prove that the roots of the complex equation $0=z^3+3z+5$ are outside the circle $S(0,1)$. I tried to guess some roots but failed. Thanks.
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### Continuous function on $[0,1]$ such that its zeros form a nowhere dense set of positive measure?

I know few facts, if $f : [0,1] \to \mathbb{R}$ is continuous, $Z(f) \triangleq f^{-1}(\{0\})$ is closed, there are continuous functions whose zeros are nowhere dense, there are nowhere dense sets of ...
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### Symmetry in the roots of a quadratic [closed]

Given $m$ is a root of $x^2+ax+b=0$ find all the possible values of $(a,b)$ such that $m^2-2$ is also a root.
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### Quintic polynomial with three real roots

I want to get a quintic polynomial $f(X) \in \mathbb{Q}[X]$ whose Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong S_5$ where $L$ is the splitting field of $f(X)$. One of strategies to get it is ...
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### Why can you use substitution to find a polynomial with roots g(a), g(b)

In a maths textbook is this statement which pertains to roots of polynomials: If an equation in x has a root x=p, and if we make a substitution u = f(x), then the resulting equation in u has a root ...
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### Roots along a homotopy

Suppose we have two functions $f,g:\mathbb{R}^n\rightarrow \mathbb{R}$. Assume the existence of unique roots to both function, i.e. $x^f,x^g\in \mathbb{R}^n$ such that $f(x^f)=0=g(x^g)$. Define the ...
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### What are the integer coeffcients of a cubic polynomial having two particular properties?

Let $f(x) = x^3 + a x^2 + b x + c$ and $g(x) = x^3 + b x^2 + c x + a\,$ where $a, b, c$ are integers and $c\neq 0\,$. Suppose that the following conditions hold: $f(1)=0$ The roots of $g(x)$ are ...
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### Complex t in the Argument of the Zeta Function

In $\zeta (s)$, $s = \sigma + ti$, where $t$ is a real number, what would become of the zeta function if $t$ were to be complex? Does there exist a proof that $\zeta (\sigma + (a+bi)i)$ ...
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### How can I find the roots of this for k?

$$ln(1-e^{-kx})(1-e^{-kx})+kxe^{-kx}=0$$ I need to find $k$ in this equation , it should be a function of $x$. Any hints on how should I do it ?
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### Let $P (x )=x^4+ax^3+bx+c=0$ and have real coefficient and have all real roots . Prove that $ab \leq 0$

Let $P (x )=x^4+ax^3+bx+c=0$ and have real coefficient and have all real roots . Prove that $ab \leq 0$ First Let the roots of this polynomial (call it P(x)) be $q,r,s,t$ By Vieta's, $a=-(q+r+s+t)$ ...
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### Find a set of polynomials whose common zero set is $\{(1, 2), (0, 5)\}$.

Find a set of polynomials $\{P_1, \dots, P_n\}$, all of whose coefficients are real numbers, whose common zero set is the given set. I know what a zero set is, but I think my confusion comes from ...
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### Quickest way to find $a^5+b^5+c^5$ given that $a+b+c=1$, $a^2+b^2+c^2=2$ and $a^3+b^3+c^3=3$

$$\text{If}\ \cases{a+b+c=1 \\ a^2+b^2+c^2=2 \\a^3+b^3+c^3=3} \text{then}\ a^5+b^5+c^5= \ ?$$ A YouTuber solved this problem recently and, though he spent some time explaining it, took over 40 ...
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### Find all $z\in{C}$ such that $z^6+(-8+8i)z^3-64i=0$. [closed]

Find all $z\in{C}$ such that $z^6+(-8+8i)z^3-64i=0$. For now I have that let $w=z^3$ such that the equation becomes a quadratic equation, $w^2+(-8+8i)w-64i=0$. Not sure what to do next; any help is ...
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### Prove that $f(x)$ and $g(x)$ do not have any roots in common.

Suppose that $a(x)f(x) +b(x)g(x) = 135$ where $a(x), b(x), f(x)$ and $g(x)$ are polynomials over $F$. Prove that $f(x)$ and $g(x)$ do not have any roots in common. Any help is appreciated; thanks!
The Problem: There are multiple "rooty" equations that can be simplified to a whole number, for example: $$\sqrt{19 + 6\sqrt{2}} - \sqrt{18} = 1$$ Because: $$\sqrt{19 + 6\sqrt{2}} - \sqrt{18} = \... 1answer 95 views ### Is Gershgorin bound of roots sharp? Gershgorin circle theorem tells that the eigenvalues of a matrix A lie in the union of the associated Gershgorin circles. A=\begin{pmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 &... 1answer 51 views ### Largest real root of a degree six characteristic polynomial I am studying the growth rate of a weighted variant of self-avoiding walks and came up with a linear recurrence in terms of the fixed weights a and b whose characteristic polynomial contains the ... 1answer 20 views ### Algorithm for determining all the zeros of a complex rational function without initial guesses Given a rational function R(x) = P(x)/Q(x), where P and Q are polynomials which can have complex coefficients, is there an algorithm which allows us to determine the zeros of R without an initial ... 3answers 687 views ### Finding number of roots using Rolle's Theorem, and depending on parameter I need to count the number of real solutions for  f(x) = 0  but I have an m in there.$$ f(x) = x^3+3x^2-mx+5  I know I need to study $m$ to get the number of roots, but I don't know where to ...
I am solving following question based on quadratic equation If $a,b,c$ are positive rational numbers such that $a>b>c$ and the quadratic equation $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root ...