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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.

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4answers
51 views

Rational Roots of a Quadratic Equation

If $a,b,c$ are non zero, unequal rational numbers then prove that the roots of the equation $$(abc^2)x^2+ 3a^2cx+b^2cx-6a^2-ab+2b^2=0 $$ are rational. Use theory of equations and basic ...
0
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1answer
55 views

Complex solutions of equation

I'm trying to find the roots of P(x) :$$x^3-3x\enspace-4=0$$ First I tried $P(\pm1),\enspace P(\pm2), \enspace P(\pm4)$ , and I found no rational solutions. So I used the formula for the solutions ...
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0answers
31 views

Prove that all the roots of the following polynomials are real [duplicate]

Suppose $p(z)$is a polynomial in $\mathbb{C}[x]$ where all of its roots lie in the lower half plane. Let $$p(z) = a_nz^{n}+a_{n-1}z^{n-1}+ _{\dots} +a_1z+a_0.$$ Consider the following polynomials $$\...
1
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2answers
37 views

The root sign and it's relation with 1/2.

$\sqrt{4}=2$ But is it same as writing... $4^{1/2}=2$? Basically I do not understand why $\sqrt{}$-sign equals $1/2$?
-1
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1answer
33 views

Consider $P(x) = 5x^6 - ax^4 - bx^3 - cx^2 - dx - 9$, where $a$; $b$; $c$; $d$ are real. If the roots of $P(x)$ are in AP, find the value of $a$. [on hold]

Consider $P(x) = 5x^6 - ax^4 - bx^3 - cx^2 - dx - 9$, where $a$; $b$; $c$; $d$ are real. If the roots of $P(x)$ are in arithmetic progression, find the value of $a$. Although I am sure that this ...
2
votes
2answers
62 views

How to find the coefficient of $a$ in a quadratic equation?

If a root of the equation $$3x^2 + 4x + 12a + 9ax = 0$$ is greater than 6, then the correct statement of the coefficient $a$ is: a) $a = 2$ b) $a> -2$ c) $a = -2$ d) $a <-2$ ...
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0answers
50 views

Are these the Big and Little Picard Theorems?

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 9.3, 9.4 Exer 9.3, Exer 9.4 (*) seem to be the Big and Little Picard Theorems or at least ...
1
vote
2answers
100 views

Proof of Casorati-Weierstrass

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch9.2 I have questions on the proof of Casorati-Weierstrass Thm 9.7 - ($\color{...
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3answers
2k views

Why can you find the roots a of polynomial by factoring it?

Let $f(x) = x^2 - 9x- 10$ We can state that $f(x) = (x + 1)(x-10)$ since I simply factored it. The roots of this function is $-1$ and $10$. However, what is the relationship between a factored ...
4
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2answers
56 views

Constant Term of a Monic Polynomial in $\mathbb{Z}[x]$ Is Divisible by 3

Given a monic polynomial $f(x)$ in $\mathbb{Z}[x]$ such that $\alpha$ and $3 \alpha$ are complex roots of $f(x),$ prove that the constant term of $f(x)$ is divisible by 3. I have attempted this ...
3
votes
1answer
41 views

Construct an analytic function which has simple zeros at all $m+in$

I found a "duplicate" question: Entire function having zeros at $m+in$, and I went through some wiki pages of Weierstrass sigma function, but they don't seem to have constructions. So basically I'm ...
1
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2answers
48 views

How many zeros does the polynomial have in the right half plane?

The polynomial is $f(z) = z^4+\sqrt{2}z^3+2z^2-5z+2$ If you check the image of the imaginary axis, you see that there are no zeros, so we can use the right semicircle from $iR$ to $-iR$,and make $R$ ...
0
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0answers
28 views

How to calculate total count of real roots of a polynomial with a higher degree than four (so the solutions can't be algebraically expressable)?

i am currently writing a lightweight library for polynomial manipulation in C and i need a function to calculate total count (not the values of them) of the real roots of a polynomial. But i couldn't ...
8
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3answers
825 views

Some interesting observations on a sum of reciprocals

This recent question is the motivation for this post. Consider the following equation $$\frac1{x-1}+\frac1{x-2}+\cdots+\frac1{x-k}=\frac1{x-k-1}$$ where $k>1$. My claims: There are $...
2
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2answers
141 views

Let $m$ be the largest real root of the equation $\frac3{x-3} + \frac5{x-5}+\frac{17}{x-17}+\frac{19}{x-19} =x^2 - 11x -4$ find $m$ [on hold]

Let $m$ be the largest real root of the equation $$\frac3{x-3} + \frac5{x-5}+\frac{17}{x-17}+\frac{19}{x-19} =x^2 - 11x -4.$$ Find $m$. do we literally add all the fractions or do we do something ...
1
vote
1answer
47 views

Is there a zeta function(with a Dirichlet series) having known roots off the critical line?

Is there a zeta function(with a Dirichlet series) having known roots off the critical line? I thought there was something like the Hilldebrand-Davis zeta function or something like that, but I can't ...
0
votes
1answer
43 views

If the roots of the equation $ax^2-2bx+c=0$ are complex ,then find the number of real roots of the equation $4e^x+(a+c)^2(x^3+x)=4b^2x$.

If the roots of the equation $ax^2-2bx+c=0$ are imaginary,then find the number of real roots of the equation $4e^x+(a+c)^2(x^3+x)=4b^2x$. The only information i'm able to interpret is$-$ Since the ...
-1
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1answer
59 views

Holomorphic $f$ has a pole $\iff f(z) = \frac{g(z)}{(z-z_0)^m}$

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch 9.2 Cor 9.6 of Prop 9.5(*) Suppose $f$ is holomorphic in $\{0<|z-z_0| < R\}$. ...
-1
votes
1answer
33 views

l is equal to the minimum value of the expression 2(y-2)^2 + 4(x -7)^2 + (y+4)^2 find [2018/l] where [] denotes the greatest integer function

$l$ is equal to the minimum value of the expression $2(y-2)^2 + 4(x -7)^2 + (y+4)^2$. Find $[2018/l]$ where $[\cdot]$ denotes the greatest integer function. Here is my approach: In order to obtain ...
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4answers
67 views

If the roots of the equation $x^2 + 3x -1 = 0$ also satisfy the equation $x^4 + ax^2 + bx + c = 0$ find the value of $a + b + 4c + 100$

If the roots of the equation $x^2 + 3x -1 = 0$ also satisfy the equation $x^4 + ax^2 + bx + c = 0$ find the value of $a + b + 4c + 100$ I tried really hard but the most I could get is the sum of the ...
0
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4answers
50 views

Let $P(x)=x^2+bx+c$, where $b$ and $c$ are integers.

Let $P(x)=x^2+bx+c$, where $b$ and $c$ are integers. If $P(x)$ is a factor of both $f(x)=x^4+6x^2+25$ and $g(x)=3x^4+4x^2+28x+5$, then $P(x)=0$ has imaginary roots $P(x)=0$ has roots of ...
2
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1answer
44 views

Find the value of all $x$ satisfying $(f\circ g\circ g\circ f)(x)=(g\circ g\circ f)(x)$, where $(f\circ g)(x)=f(g(x))$.

Let $f(x)=x^2$ and $g(x)=\sin(x)\ \forall\ x\in \mathbb R$. Then find the value of all $x$ satisfying $(f\circ g\circ g\circ f)(x)=(g\circ g\circ f)(x)$, where $(f\circ g)(x)=f(g(x))$. Solution. ...
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9answers
57 views

Common root question on quadratics equations to show that $a+b+c=0$

If $f(x)=ax^2+bx-c$ and $g(x)=ax^2+cx+b$ have a common root , show that $a+b+c=0$. I tried this by thinking that $\alpha$ is the common root and then I got by substituting and solving , $(b^2+c^2)(b-...
1
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1answer
42 views

Descartes principal imaginary roots

If $\ a_1,\ a_2,\ a_3,...\ a_n$ $n\ge2$are real and $(n-1)\ a_1^2 -2n\ a_2 <0$ prove that at least two roots of the equation $x^n+\ a_1x^{n-1}+\ a_2x^{n-2}+...+\ a_n=0$ are imaginary. Manually ...
1
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1answer
71 views

Can we prove that all roots of those polynomials are real negative?

Suppose I have a strictly decreasing sequence of positive constants: $$ K = \{K_1, \dots, K_n \, | K_i \in (0;1) \wedge K_i > K_{i+1} \} $$ And I create polynomials as follow (I read they are ...
0
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2answers
75 views

Properties of roots

I've been struggling to understand the procedure shown below. Somehow, by setting x=0 I managed to get to the equation (1.12). However, I think didn't get it really as I do not know how to compare the ...
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2answers
46 views

$f(z_0)=w_0$ Then $\exists \epsilon, \delta$ such that $f(z)-\alpha$ has exactly $m$ simple roots in $B(z_0,\delta)$ for $|\alpha-w_0|<\epsilon$

Let $f$ be analytic at $z_0 \in \mathbb{C}$ and $f(z_0)=w_0$. Suppose that $f(z)-w_0$ has a zero of finite order in $m \geq 1$ at $z=z_0$. We need to show that there exist $\epsilon >0$ and $\...
2
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2answers
71 views

Number of solution of $x^4-5x^3+(\lambda+2)x^2-5x+1=0$

Consider the bi-quadratic equation $E:x^4-5x^3+(\lambda+2)x^2-5x+1=0$ then, the real values of $\lambda$ so that $E$ has four different solutions is? My attempts: As $x=0$ is not a solution for any $...
1
vote
1answer
78 views

Integral of $\sqrt{4 - x^2}$? [duplicate]

So, I've been struggling to solve the following exercise: $$ \int_0^1 \sqrt{4-x^2}\,dx $$ Most of the solutions I've seen online use substitution using $x = \cos(t)$ or $x = \sin(t)$, however I'm ...
0
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1answer
43 views

Are roots of $\det(A-tB)$ with symmetric matrices real? [duplicate]

Let $A$, $B$ be symmetric real matrices $n \times n$-type. Let's consider the polynomial $$ f(t)=\det(A-tB). $$ If $B$ is positive definite, then it is known that $f$ has only real roots. Is the same ...
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2answers
55 views

Find an integer $a$ such that $(x-a)(x-10)+1=(x-b)(x-c)$ for some integers $b$ and $c$ [closed]

Can someone help with this Olympiad question? Find an integer $a$ such that $$(x-a) (x-10) +1$$ can be factored as $$(x-b) (x-c)$$ with $b$ and $c$ integer.
4
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2answers
105 views

Numerically find cubic polynomial roots where coefficients widely vary in magnitude

Consider the following polynomial: $$ p(x) = x^3 + (C_b+K_a)x^2 - (C_aK_a + K_w)x - K_aK_w $$ Where: $x, C_a, C_b$ are concentrations, positive real numbers typically within $[10^{-7};1]$. The ...
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0answers
52 views

Zeros of improper integral

Let $f(x) = \int_0^\infty g(x,t)dt$, then how can we find the number of zeros (or an upper bound for the number of zeros) of f(x)? For example, What about $f(x)=\int_0^\infty e^{-t}t^{x}\sum_1^n ...
1
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2answers
58 views

Prove that sufficiently large partial sums of the Taylor series expansion of $e^z$ have all roots outside of an arbitrarily large radius.

Consider the complex-valued family of functions $$ f_n(z) = \sum_{k=0}^n \frac{1}{k!}z^k. $$ Is it possible to use Rouché's Theorem to prove that for any $R \in \mathbb R$ there exists some $n$ ...
1
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1answer
44 views

What's the intuition behind the fact that you can factorise an expression as $(x-a)(x-b)$ if $a$ and $b$ are the roots of the expression

Say $f(x)$ is a polynomial function with roots $a, \, b, \, c$ then this can be expressed as $f(x) = (x-a)(x-b)(x-c)$ What's the intuition behind this? Why is this true?
6
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1answer
207 views

Common roots of recursive defined polynomial

I have a series of polynomials $P_j(x)$ given by the recursive formula $$P_{j+1}=\frac{e_j}{c_j}xP_{j}-\frac{f_j}{c_j}P_{j-1} $$ with $P_{-1} \equiv 0$, $P_0 \equiv 1$, where $$c_j = (j+1)(j+2\kappa+1)...
0
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1answer
25 views

Negative square roots of Reals in inequalities

Where r $ \epsilon \Bbb { R,}$ determine the range of r. $$\left\lvert 1-2 \sqrt{-r}\right\rvert < 1$$ $$-1< (1-2 \sqrt{-r}) < 1$$ $$-2< (-2\sqrt -r) < 0 $$ $$1>(\sqrt -r) > ...
6
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2answers
54 views

Roots of $f$ and $f'$ for $1+\sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)(x-2)\cdots (x-k)$

I have this question from an admission exams. Given that $$f(x)=1+\sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)(x-2)\cdots (x-k)$$ find $S(f(x))-S(f'(x))$ where $S$ denotes the sum of the real roots ...
3
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1answer
33 views

Asymptotics Neumann Solution Stefan problem

Consider the one-phase Stefan problem, defined on the moving domain $[0,s(t)]$ where the temperature inside the domain is determined by the heat equation $$T_t=T_{xx},\qquad 0<x<s(t),$$ subject ...
0
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2answers
46 views

Integer Difference Between Roots

Are there two numbers, x and y, such that neither is a perfect square, and the difference of their roots is an integer? Can you find x and y such that the difference gets arbitrarily close? What ...
1
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1answer
65 views

System of equations with special substitution

Solve the following system of equations on $\mathbb R$ \begin{cases} \dfrac{x+y}{1+xy}= \dfrac{1-2y}{2-y}\\[6px] \dfrac{x-y}{1-xy}=\dfrac{ 1-3x}{3-x} \end{cases} Solution Setting $ \begin{cases} x=...
7
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0answers
136 views

Does $N_0(T) =N(T)$ where $N$ is the exact Riemann $\zeta$ zero counting function and $N_0$ is the approximate zero counting function imply the RH?

From A theory for the zeros of Riemann ζ and other L-functions The main new result is that the zeros on the critical line are in one-to- one correspondence with the zeros of the cosine ...
2
votes
3answers
76 views

Calculate $\lim_{n\to\infty} \sqrt{5n^2+4}~-~\sqrt{5n^2+n}$

While working on limits in math my teacher gave us limits involving square roots as the following Calculate:$$\lim_{n\to\infty} \sqrt{5n^2+4}~-~\sqrt{5n^2+n}$$ and even though I know how to handle ...
1
vote
1answer
48 views

Solving $\sqrt{\frac{\log\left(x + a + n\right)}{a}} - \sqrt{\frac{\log\left(x + a + n\right)}{x}} = \Phi$ for $x$ [closed]

Consider the following equation: $$\sqrt{\frac{\log\left(x + a + n\right)}{a}} - \sqrt{\frac{\log\left(x + a + n\right)}{x}} = \Phi$$ where all the variables belong to $\mathbb{R}$, and: $\log$ is ...
2
votes
4answers
174 views

Relevance of Complex roots of Quadratic Equation

Let's say Amy is a stunt pilot, planning on doing a parabolic dive in an air show: $y = x^2 + 4x +5$ She hopes to use this trajectory to dive close to the ground (the x-axis, height is the y-axis), ...
1
vote
2answers
54 views

Compare largest root of two polynomials

For $1\leq a, a+1<b$ with $a,b\in\mathbb{N}$ let $f:=f(a,b):=\sum_{i=2}^{b-a}i,$ $g:=g(a,b):=f(a,b)+a+b+1$ $h:=h(a,b):=f(a,b)-(b-a-1)$ and consider the polynomial $$ p(t):=t^g-\left(\sum_{i=h}^f ...
0
votes
0answers
36 views

Solving $\frac{(\sqrt{x} - \sqrt{a})^{2}}{a x} \cdot \log\left(x + a + n\right) = \psi$ for $x$

Consider the following equation in $\mathbb{R}$: $$\frac{\left(\sqrt{x} - \sqrt{a}\right)^{2}}{ax} \cdot \log\left(x + a + n\right) = \psi$$ for $x$, $a$, $n$ and $\psi$ strictly positive, and where ...
3
votes
7answers
108 views

The number of real roots of $x^5 - 5x + 2 =0$

How many real roots does the equation $x^5 - 5x + 2 =0$ have? I know the following facts: The equation will have odd number of real root. That function cannot have rational root. The function will ...
7
votes
3answers
137 views

Proof involving polynomial roots

From USAMO 1977: "If $a$ and $b$ are two roots of $x^4+x^3-1=0$, prove that $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$." The solutions I've seen for this problem all involve manipulating Vieta's ...
4
votes
0answers
79 views

Finding zeros of function by integration: a novel relationship or not?

It seems that in certain cases one can find the zero of a function by solving an integration problem instead. This surprises me, and I am wondering to what extent this (1) has been studied, and/or (2) ...