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

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1answer
303 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
2
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1answer
54 views

Solve the equation for $X$

$$X^3-3X^2+3X=\frac{3R-10}{2}$$ How can i solve it for $X$ ? I tried to do : $$\Rightarrow X(X^2-3X+3)=\frac{3R-10}{2}$$ ???
4
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0answers
246 views

Determine the number of zero points of $z^8-5z^3+z-2$ within the open unit circle (Rouché?)

How many zero points does the polynomial $z^8-5z^3+z-2$ have within the open unit circle? Hello, consider $$ \gamma\colon [0,2\pi]\to\mathbb{C}, \varphi\longmapsto\exp(i\varphi) $$ and ...
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3answers
350 views

Finding the zeros of $f(x)=-x^3-x^2+7x+7$

$$f(x)=-x^3-x^2+7x+7$$ it needs to be solved for the zeros I need to figure out the answer to this please help I have tried many different things and I'm confused
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6answers
621 views

How do I solve and plot the complex equation

I have the following complex equation: \begin{equation} z^6 + 1 = 0 \end{equation} I would like to be able to gain some intuition and understanding. I know from the fundamental theorem of algebra ...
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4answers
346 views

How prove this $\displaystyle\lim_{n\to \infty}\frac{n}{\ln{(\ln{n}})}\left(1-a_{n}-\frac{n}{\ln{n}}\right)=-1$

let equation $x^n+x=1$ have positive root $a_{n}$. show that $$\displaystyle\lim_{n\to \infty}\dfrac{n}{\ln{(\ln{n}})}\left(1-a_{n}-\dfrac{n}{\ln{n}}\right)=-1$$ some hours ago,it prove that ...
3
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1answer
120 views

How prove this limit $\displaystyle\lim_{n\to \infty}\frac{n}{\ln{n}}(1-a_{n})=1$

let equation $x^n+x=1$ have positive $a_{n}$. show that $$\displaystyle\lim_{n\to \infty}\dfrac{n}{\ln{n}}(1-a_{n})=1$$ yesteday, I have post this and prove following $$\displaystyle\lim_{n\to ...
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2answers
48 views

Help for two values of x

I'm looking for help. Even if you just tell me the process rather than the answer. Given that $y=10-3x^2$, find two values of $x$ for which $y=-17$. How would I go about answering this?
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3answers
547 views

How prove this limit $\displaystyle\lim_{n\to \infty}a_{n}=1$

Let $a_n$ be the only positive root of the equation $x^n+x=1$, for each $n\in \Bbb N$. Show that $\lim \limits_{n\to \infty}a_{n}$ exists,and find its value. My guess is that $$\lim \limits_{n\to ...
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1answer
398 views

Roots of biquadratic equation

This question also was a part of my today's maths olympiad paper: If squares of the roots of $x^4 + bx^2 + cx + d = 0$ are $\alpha, \beta, \gamma, \delta$ then prove that: $64\alpha\beta\gamma\delta ...
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3answers
84 views

All the roots of $(x^2+1)^2 = x(3x^2+4x+3)$

Find all the roots of the equation : $$(x^2+1)^2 = x(3x^2+4x+3)$$How do we find the roots in polynomials of degree > 2 ?? Also, In odd degree polynomials i use descartes rule of signs to predict the ...
3
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1answer
253 views

Solve $x+y+z = x^3 + y^3 + z^3 = 8$ in $\mathbb{Z}$

Solve $x+y+z = x^3 + y^3 + z^3 = 8$ in $\mathbb{Z}$ First I tried to transform this equation, substituting $x = 8-y-z$. So I end up with: $$x^3 + y^3 + z^3 = 8$$ $$(8-y-z)^3 + y^3 + z^3 = 8$$ ...
3
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2answers
2k views

Finding the discriminant and roots of a polynomial

How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here $\Delta$ is the ...
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5answers
254 views

Don't know how to find all the roots

So i got this problem : Find all the roots of $r^{3}=(-1)$ i can only think to use : $\sqrt[n]{z} =\sqrt[n]{r}\left[\cos \left(\dfrac{\theta + 2\pi{k}}{n}\right) + i \sin\left(\dfrac{\theta + 2\pi ...
1
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1answer
179 views

Polynomial Functions - Rational Root Theorem to find Zeros

I apologize if the level of this question is too low for this forum, it's my first time posting. I was reading about how to find the zeroes of a polynomial function, and I came across using the ...
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2answers
323 views

Determine the number of zeros of the polynomial $f(z)=z^{3}-2z-3$ in the region $A= \{ z : \Re(z) > 0, |\Im(z)| < \Re(z) \}$

Question: a). Determine the number of zeros of the polynomial $$f(z)=z^{3}-2z-3$$ in the region $$A= \{ z : Re(z) > 0, |Im(z)| < Re(z) \}$$. (b). Find the number of zeros of the function ...
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0answers
38 views

Counting roots of sums of sigmoids

Let $f(x)=\sum_i a_i\tanh(b_ix+c_i)+d$ be the class of sums of $n$ sigmoids parameterized by $a,b,c,$ and $d$, with all values being real. I suspect, but can't prove, that the number of roots of $f$ ...
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1answer
113 views

The real roots of $(x-41)^{49}+(x-49)^{41}+(x-2009)^{2009}$

What all concepts should I know to answer this question? Just give the basic guidelines and then, I will try to solve.
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0answers
57 views

Roots of A Non-linear Equation

I have the following non-linear equation $$b_1\left(\frac{1}{f_1^2}-\frac{1}{(f_1-a_1)^2}\right)=b_2\left(\frac{1}{f_2^2}-\frac{1}{(f_2-a_2)^2}\right)$$ where $$f_1+f_2=A(\ \mbox{constant})$$ When I ...
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3answers
751 views

For which values of $k$ will $ x^3 -x^2 -8x +k = 0$have 3 real roots?

I have the following equation: $$ x^3 -x^2 -8x +k = 0$$ The question: For which values of $k$ will the cubic equation have 3 real roots? Thank you
2
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2answers
94 views

$a+b\sqrt{2}$ not a root of monic polynomial over $\mathbb{Z}$

Consider $a+b\sqrt{2}$ for $a,b \in \mathbb{Q}-\mathbb{Z}$ . I need to show that it cannot be a root of any monic polynomial with coefficients in $\mathbb{Z}$
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1answer
51 views

Is the case where the zeros of $f$ or $g$ are isolated possible? [closed]

Assume that $f,g:\mathbb{C}→\mathbb{R}$. Let us consider the following equation in $\mathbb{C}$ $$f(s)g(s)=0$$ My question is: What are the cases where the zeros of $f$ or $g$ are isolated?
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2answers
16k views

Quadratic equation - Alpha and Beta Roots

If α and β are the roots of the equation x² + 8x - 5 = 0, find the quadratic equation whose roots are α/β and β/α. My working out so far: I know that α+β = -8 and αβ = -5 (from the roots) and then i ...
26
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1answer
479 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
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1answer
102 views

At most one positive root for a sum of fractions

It seems that the following equation has at most one positive root $x \in \mathbb{R}$. How should I approach to prove it? $\sum_{i=1}^K \frac{1}{x+id} = \sum_{i=1}^N \frac{1}{x+i}$ where $K < N$ ...
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0answers
251 views

Cubic roots and Cardano formula

On solving the cubic equations, applying Cardano formula yield complex results. I wanted to evaluate the exact roots (not numerical) but I ended up with complex numbers/nested radicals. To get rid ...
2
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1answer
214 views

Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...
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0answers
22 views

Dependence on Parameters of the Solution of a Non-linear Equation

I have the following equation for the delay in a queue\begin{align} d(f)=\frac{c(1-f)^2}{2(1-a)}+\frac{\lambda b}{2f(f-a)}\end{align} where $0\le f\le 1;\quad c,a=\lambda\tau, \ b=\tau^2$ or ...
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2answers
119 views

Formula for roots of polynomials

For a quadratic polynomial there exists a formula for its roots. I read that similarly for polynomials of degree 3 and 4 there also exists such a formula but that no such formulas exist for ...
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1answer
197 views

finding value of constant such that function has distinct root

we have the function cubic function $$ x^3 -12x +k =0 $$ it has distinct root in $$ [0,2{]} $$ that task given to us is to find the the value of k satisfying the above conditions I proceeded ...
12
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2answers
122 views

Fully factored integer polynomials with constant differences

Given a degree $d$, it is possible to construct a pair $(F,\delta),$ where $F$ is a polynomial in $\mathbb{Z}[X]$ and $\delta$ a non-zero integer, such that $F(X)$ and $F(X)+\delta$ both split into ...
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2answers
123 views

verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
3
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1answer
243 views

how to find the roots of a cubic equation?

Given a formula $$x^3+ax^2+bx+c=0$$ how can I get the value of x without having an $i$ in my roots? Because Cardano's formula does have imaginary numbers if the discriminant is less than zero. My ...
7
votes
4answers
529 views

How many zeros does $z^{4}+z^{3}+4z^{2}+2z+3$ have in the first quadrant?

Let $f(z) = z^{4}+z^{3}+4z^{2}+2z+3$. I know that $f$ has no real roots and no purely imaginary roots. The number of zeros of $f(z)$ in the first quadrant is $\frac{1}{2\pi ...
0
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1answer
66 views

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if a>0 and b<0

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if $a>0$ and $b<0$. How do i find the value? This doesn't make any sense.
2
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0answers
60 views

Roots of a polynomial plus a logistic equation

I would like to know if there are any methods to find the roots (analytically) of complex valued equations of the following form: $$ f(z)=P(z)+\frac{e^{-z}}{(1+e^{-z})^2} $$ where $P(z)$ is a ...
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2answers
630 views

How to factorize $x^3 - 7x + 6$?

How do you factorize this polynomial: $\mathbf{x^3 - 7x + 6}$ Some online solver doesn't even work saying: using GCF method doesn't work, but sites like Mathway.com gave me the answer, is there a ...
1
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1answer
54 views

What's a good reference for the continuity of the number of zeros?

What is a good reference for the following statement, or something that easily implies it? For all sequences $\:\langle\:f_0,f_1,f_2,f_3,...\rangle\:$ of (complex) analytic functions $\:\:f_n : ...
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2answers
97 views

What this sine function equation means?

Apostol's book "Calculus" asks to prove that $$\sin\frac{\pi }{6}=\frac{1}{2}$$ using the fact that $$\sin 3x=3\sin x-4\sin^3 x$$ and $$\sin \frac{\pi}{2}=1$$ So, we take $x=\frac{\pi}{6}$ and ...
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2answers
118 views

Prove that if a polynomial $P$ has no roots in the upper half plane, then so does $P'$

Prove that if a polynomial $P$ has no roots in the upper half plane, then so does $P'$ This is a part of an exam preparation and I would appreciate a hint. My approach was to use Rouche's theorem but ...
1
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3answers
377 views

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots.

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots. MY attempt: We can open and get a bi quadratic but that is two difficult to show that it has real roots.THere must be an ...
6
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2answers
485 views

Show that a polynomial has at least 1 real root

I have the polynomial $P(x)=x^{2}+2013x+1$ and a number $n\in\mathbb{N}$. Now I have to show that the polynomial $P(P(...P(x)...)$ $(n$ times$)$ has at least one real root. How can I do this?
4
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1answer
35 views

How does the set of algebraic numbers compare to the set of possible fixed points for polynomials (with integer coefficients but not y=x)?

I was thinking of a way to map any polynomial $P$ with at least one real root onto some polynomial $Q$, s.t. the real roots of $P$ are exactly the real fixed points of $Q$, (There could be many, so we ...
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2answers
707 views

$n,a\in \mathbb Z,n\geq1,$ prove that $x^3+x+1\nmid x^n+a$

$n,a\in \mathbb Z,n\geq1,$ prove that $x^3+x+1\nmid x^n+a.$ In other word, they have no common roots. My idea: Let $x_1,x_2,x_3$ be the roots of $x^3+x+1=0,$ we need to prove that $\dfrac{x_1}{x_2}$ ...
12
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2answers
418 views

Number of real positive roots of a polynomial?

Consider the polynomial $$f(x)=x((1+x^n)^n+a^n)-a(1+x^n)^n,$$ where $n\geq 2$ is a positive integer and $a$ is a positive real number. I'm interesting in deducing the number of positive real roots ...
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3answers
130 views

How to prove that a given polynomial $P(x)$ has no interger roots.

How to prove that a given polynomial $P(x)$ has no integer roots.
5
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3answers
778 views

How to solve problems involving roots. $\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$

How to solve problems involving roots. If we square them they may go to fourth degree.There must be some technique to solve this. $$\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$$
0
votes
3answers
92 views

Find the eigenvalues of the matrices.

The characteristic equations for the two matrices are: $x^3-8x-7=0$ and $x^3-6x^2+11x-6=0$ I know that in order to find the eigenvalues, I need to factor these two equations out. I'm just having a ...
0
votes
2answers
66 views

Root of an exponential equation

Let $0 \le a \le 1$ and $-\infty < b < \infty$. I am looking for a solution of the exponential equation. $$ a^x + abx = 0. $$ I guess closed form expression of the root in terms of $a$ and $b$ ...
1
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2answers
346 views

what is the maximum number of roots of quadratic function with 3 variables?

Given the general quadratic form with $3$ variables $(x,y,z):ax^2 + by^2 +cz^2 + dxy + eyz +fzx + gx + hy + iz = 0$ satisfies $x^2 + y^2 + z^2 = 1$ I would like to ask what is the maximum number of ...