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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2answers
52 views

Number of real roots of $2^x = 1-x^2$ for $x\in (0,1)$

How can I found no. of real roots of $2^x = 1-x^2$ in $x\in (0,1)$ I did not found a method by which i can draw graph of two curve in the interval $x\in (0,1)$ please help me , Thanks Sorry ...
10
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3answers
298 views

Do there exist an infinite number of complex solutions of $3^z+4^z=5^z$?

Are the followings true? 1 : There exist an infinite number of complex solutions of the equation $$3^z+4^z=5^z \tag{$\star$}.$$ 2 : There exist an infinite number of complex solutions $z$ of ...
7
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2answers
194 views

Square root of a natural number to square root of another natural number

Is there such integers $x,y$ which they're not perfect squares and they're not equal, such that: $\sqrt{x}^\sqrt{y}$ is actually an integer? Or rational number?
2
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3answers
122 views

If I know that a polynomial (of order $k \gt 2$) has at most $1$ positive real root - can I find that easily?

[update 2] Urgghh - the time-consumption really stems only from the construction of the h-order polynomial. The time for finding the roots (only 10 to 20 times Newton-iteration because of my nice ...
4
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3answers
133 views

Show that the real part of the root of an equation is constant

I've been stuck for a while on the following question: Let $z$ be a root of the following equation: $$z^n + (z+1)^n = 0$$ where $n$ is any positive integer. Show that $$Re(z) = -\frac12$$ ...
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3answers
1k views

Determine the number of real roots in the equation…

Determine the number of real roots in the equation $2x^3 + x^2 = 3$. I know about finding the different roots, and solving giving that it has (for example) 2i as a root, but I'm not sure how to just ...
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0answers
131 views

How prove this $f^{(n)}(x)=0$ has at least $n-1$ distinct roots

This question is from Mathematical Analysis I(Zorich) Page 232 problem 6(c), let $f\in C^{(n)}\left(]-1,1[\right)$ and $\displaystyle\sup_{-1<x<1}|f(x)|\le 1$.let $m_{k}(I)=\inf_{x\in ...
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1answer
52 views

Trying to find the root of the derivative of the MLE for a simple linear regression model

I have a function $$l(\beta_0, \beta_1, \sigma^2) = -\frac{n \log(2\pi)}{2} - n \log \sigma - \frac{1}{2 \sigma^2} \sum_{i=1}^{n} (y-\beta_0 - \beta_1 x_i)^2$$ which is the log-likelihood function of ...
3
votes
1answer
139 views

A polynomial has only real roots and all coefficients $\pm 1$. Prove the degree $<4$.

Let $P(x)$ be a polynomial with only real roots and all coefficients equal to $\pm 1$. Prove that the degree of the polynomial is less than 4. This is practice for Putnam, but I am not certain where ...
3
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2answers
182 views

Analytical solution of a polynomial with non integer order

Can anyone think of a possible analytical solution of the following equation? $x\left(1-0.2x^2\right)^{5/2}=constant$ I am not a mathematician, but, it seems to me that only numerical methods can ...
3
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2answers
144 views

Find roots of polynomial with degree $\ge 5$

During our research we came up with the problem of computing the root of a polynomial of degree $\ge 5$ exactly. The coefficients are, except for the linear and constant term, all non-negative and ...
0
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1answer
113 views

Help with finding the roots of a function

I have the following problem: Find the smallest root of the function $e^{-x} = \sin (x)$ and focus the root with Newton's method to $8$ decimal accuracy. Any suggestions? :) Thank you for any ...
133
votes
1answer
4k views

Rational roots of polynomials

Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ has exactly $n$ distinct rational ...
1
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4answers
165 views

Eigenvalues and the Characteristic Equation

Given the following matrix, $$ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} $$ assuming eigenvectors exist for $A$, they can be found by first solving for $\lambda$ ...
2
votes
3answers
83 views

In a numerical system of base $r$, the polynomial $x^2 − 11x + 22 = 0$ has the solutions $3$ and $6$. What is the base r of the system?

From Algebra, the statement is equivalent to say that $(x^2− 11x + 22)_{r}$ = $(x − 3)_{r} \cdot (x − 6)_{r}$. Doing operations we arrive at $3 + 6 = 11_{r} = r + 1$, and $(3)(6) = 22_{r} = 2 \cdot ...
2
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1answer
45 views

Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a ...
1
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1answer
113 views

Condition on coefficients of polynomial to guarantee special zero-distribution

Consider polynomial below: $P(s) = a_4s^4 + a_{3}s^{3} + a_2s^2 + a_1s + a_0 $ The question is: Under which conditions on coefficient of above polynomial, we claim that the zeros of polynomial ...
0
votes
2answers
302 views

Newtons method and finding stationary points

I have an equation $l(x) = \sqrt{(x - 0.2)^2 + (x^2 - 2.7)^2}$. Now I basically want to find at which x coordinate that $l(x)$ will be it's smallest. I have differentiated the equation to find ...
1
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1answer
324 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
votes
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
311 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 ...
1
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3answers
357 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
771 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 ...
7
votes
4answers
352 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
votes
1answer
121 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 ...
1
vote
2answers
49 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?
8
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3answers
569 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 ...
1
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1answer
472 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 ...
3
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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
votes
1answer
261 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
votes
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 ...
1
vote
5answers
255 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
vote
1answer
207 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 ...
3
votes
2answers
371 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 ...
1
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0answers
39 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 $f$ ...
2
votes
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.
1
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0answers
58 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 ...
0
votes
3answers
800 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
votes
2answers
95 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 $\mathbb{Z}$
0
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1answer
53 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?
3
votes
2answers
19k 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
494 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 $K < N$ ...
2
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0answers
264 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
votes
1answer
248 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 ...
0
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0answers
23 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 ...
3
votes
2answers
129 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 ...
1
vote
1answer
217 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
votes
2answers
125 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 ...
1
vote
2answers
126 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 ...