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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1answer
215 views

Radical solution to a polynomial quartic equation

Consider the following quartic equation: $$x^4 + rx^3 + r^2x^2 + r^3x + r^4 - 1 = 0$$ By Lodovico Ferrari solution, this equation must possess four radical solution provided that $r$ is a rational ...
3
votes
3answers
57 views

Zeroes of polynomials and their sum

Let $a, b$ are zeroes of the polynomial $x^2-10cx-11d$ and $c,d$ are the zeroes of the polynomial $x^2-10a x-11b $ where $a,b,c, d$ are distinct reals then $a+b+c+d=?$
4
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0answers
108 views

Roots of a polynomial

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then $p_n(a)\...
0
votes
0answers
36 views

What will happen if there is a way predicting at a least one root of $p_{n}(x)=0$ without calculator?

let $p_{n}(x)$ be a polynomial of degree $n$ defined as follow : $p_{n}(x)=x^n +a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....a_{0}$ which : $a_{n-1},a_{n-2},.....,a_{0}$ are non nul real numbers coefficients. ...
2
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2answers
56 views

Concerning Roots of the cubic equation $f(x)=x^3+x^2-5x-1$ and the Greatest Integer (or Floor) function

The Question I got into a rather tight corner with this question. It says: Let $\alpha, \beta, \gamma$ be the roots of $f(x)=0$, where $f(x)=x^3+x^2-5x-1$. Then, the value of $\lfloor\alpha\...
3
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0answers
63 views

Sum of zeros of $P(x)$

I asked this question here before too, but vaguely, hopefully, this time will be a better attempt: There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero ...
1
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1answer
71 views

Finding the real root of the polynomial $2x^3-3x^2+2 $

I want to get exactly roots of this equation... $2x^3-3x^2+2 = 0$ I try to solve it but can not find the solution. wolframealpha just give me aproximation.. I know the real root is $-1< root &...
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1answer
137 views

How can I prove this statement about square root?

Introduction In computer science there is a field called Formal Methods and Specifications. In this field software designers design softwares by specifying their functionalities in formal methods, ...
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3answers
164 views

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then $a=b=c=0$ [closed]

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then it is an identity i.e. it is true for all values of $x$ and $a=b=c=0$. What is a proof of this?
1
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1answer
58 views

Can we write “fractional root” symbol in math?

Fractional exponents are legit but I have never seen fractional roots, so I just wonder if we can write fractional roots such as this: It sometimes can be convenient to think about too.
20
votes
3answers
562 views

Probability of $ax^2 + bx + c = 0$ having real solutions

$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real solutions? ...
0
votes
0answers
59 views

Finding the roots of weighted sum of $\tanh$

How can this equation be solved analytically? Is it possible, why? Is there a better alternative than the Newton approximation? $k\tanh^2(ax+b) + l\tanh^2(cx+d) + m = 0 $ with $a,b,c,d,k,l,m \text{ ...
3
votes
6answers
95 views

Solving $e^\frac1x = x$ non-graphically?

This question has come up twice in different tests and the instructions always point out that it should be solved using a graphic calculator. Fair enough, the answer is ≈ 1.76322...(goes on forever?). ...
7
votes
1answer
271 views

Solving a special Quartic Equation.

Solve for $x$ $$(x^2-4)(x^2-2x)=2$$ I have tried the Rational Root Theorem and found that there are no rational roots. Further, the polynomial $p(x)=(x^2-4)(x^2-2x)-2$ is irreducible since when ...
1
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1answer
41 views

Show that a real polynomial of degree more than or equal to $3$ is reducible

Let $f \in \mathbb R [x]$ and suppose that $\deg(f) \geq 3$. Then $f$ is reducible. Proof: By the Fundamental theorem of algebra there are $\lambda _j \in \mathbb C$ such that $$f(x) = (x-\lambda_1) \...
1
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1answer
29 views

existence of roots for a general function over the complex plane

For some more general functions other than polynomials, are there any fixed conditions for the existence of roots in a general sense? For instance, function like $z\mathrm{sin}z-1$
6
votes
4answers
242 views

Can $\sin(\pi/25)$ be expressed in radicals

I suspect that sin(pi/25) is not expressible in elementary forms in radicals because it is the root of some quintic (or rather cos(pi/25) is). Can anyone prove that that particular quintic has no ...
0
votes
1answer
40 views

How to find out if a sequence with exponentiation in fraction is convergent

I need to find the convergence of this function: $\sum^{\infty}_{x=1}{\frac{(x+1)^{x^2}}{x^{x^2}2^x}}$ Now my problem is, I have no clue how to do this (I tried the root-test and it did not work out)....
3
votes
4answers
434 views

Solving equations with exponentials and a non-exponential term.

I know how to solve exponential equations. Just use logarithms, e.g., $$ 2^x-3=0 \\ 2^x=3 \\ x=log_23 \\ $$ But on a recent math test I found an equation of the form: $$ 2^{n-3}=\frac {20}{n} $$ ...
5
votes
4answers
110 views

Root of $(x+a)^{x+a}=x^{x+2a}$ and $e$

Let us denote solution to the equation $$(x+a)^{x+a}=x^{x+2a}$$ with $X_a$. ($a$ is a non-zero real number) Prove that: $$\lim_ {a \to 0} X_a = e$$ This is something that I ...
5
votes
1answer
148 views

Roots of a Cubic Polynomial with Elementary Symmetric Polynomial Coefficients

Let $R_n$ be a set of $n$ distinct nonzero rational numbers. Let $e_k$ be elementary symmetric polynomials over $R_n$---i.e. $e_0=1$, $e_1 = \sum_{1\le i\le n} r_i$, $e_2 = \sum_{1\le i<j\le n} r_i ...
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2answers
300 views

How to find the positive and negative roots of a function?

Iam trying to solve the following question: Find all numbers $a$, such that the equation $x^2-ax-a = 0$ has one positive root and one negative root. I've tried it already but I cannot seem to ...
1
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1answer
55 views

Trigonometric root of a polynomial

If $4\cos^2 \left(\dfrac{k\pi}{j}\right)$ is the greatest root of the equation $$x^3-7x^2+14x-7=0$$ where $\gcd(k,j)=1$ Evaluate $k+j$ I tried factorizing the equation but it wasn't of ...
3
votes
3answers
135 views

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots.

Prove $f=1+x+x^2+x^3+\cdots+x^n$ has no multiple roots. My attempt: Consider the polynomial $g=(x-1)(1+x+x^2+x^3+\cdots+x^n)$ As $f\mid g, g$ all the roots of $f$ are roots of $g$. This means I ...
0
votes
3answers
98 views

What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots…

I found this question in a maths-group in Facebook- What are the values of $p$ so that equation $x^3+(p-2)x^2+(5-2p)x-10=0$ has exactly $2$ real roots........ I think we do not count repeated roots ...
4
votes
1answer
121 views

Number of complex roots of a degree 6 polynomial

Given some degree 6 polynomial $f(x) \in \mathbb{Q}[x]$, is there any invariant of the polynomial (depending on the coefficents) that will tell you if this polynomial has 6 complex roots or just 2 ...
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6answers
55 views

AS Maths simplify

A question in my AS Exam this morning was simply, Simplify $(5\sqrt5)^3$ I tried $(5\sqrt5 \cdot 5\sqrt5)^2$ and ended with $5^4$. Is that correct? I think it's completely wrong, but it'd be awesome ...
1
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0answers
51 views

Two-term asymptotic approximation for roots of a polynomial (dominant balance)

I'm trying to find the roots to the following equation: $t^5 - \epsilon t^3 + \epsilon^3 = 0$ as $\epsilon \rightarrow 0$. From expansion $t= \epsilon^{\alpha}t_1 + \epsilon^{2\alpha}t_2 + \...
2
votes
2answers
36 views

Zeros of weighted sum of two Bessel functions

Just a simple and very tentative query to alleviate my seemingly futile internet digging: is there anything known on the structure of the entire function given by \begin{equation} f_c(z):=zJ_0'(...
1
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3answers
94 views

Distinct roots of $z^n-z$

How would we prove that for any positive integer $n$ the complex roots of $z^n-z$ are all distinct? In the case that $n=1,2,3$ I have factored it directly but how can we do it in general?
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0answers
30 views

Roots of serie $\sum_{i=1}^N \frac{a_i + b_i \log(c_i - x)}{c_i - x}$

I encountered the following serie during an optimization problem: $\sum_{i=1}^N \frac{a_i + b_i \log(c_i - x)}{c_i - x}$ I need to find the roots of that serie, but I cannot find the way to find a ...
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1answer
34 views

Prove that the line integral on $\beta$ of $f'(z)/f(z) = (A-B)/2 \pi i$ using Rouche's Theorem

Suppose that $\alpha$ is a regular closed contour. $f$, our function, lacks zeros and poles on $\beta$ and if A=the number of zeros of f inside $\beta$ (a zero of order n is counted n times) and B= ...
0
votes
1answer
29 views

If $p>0$ demonstrate that the $1/2\pi i$ the line integral of $z^p f'(z)/f(z)$ is $\sum (z_k)^p$

This is basically a deviation of Rouche's Theorem from what I can tell. My first instinct was to do this via induction in which we know that $p=0$ we would have Rouche's theorem. But it gets ...
2
votes
1answer
70 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| \...
0
votes
1answer
98 views

Laguerre theorem

I'm looking for a proof of the theorem 7, page 6, of this document : http://www.nipne.ro/rjp/2013_58_9-10/1428_1435.pdf Theorem 7 (E. Laguerre) Let $f \in \mathbb{R}[x]$ be a polynomial of degree ...
1
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0answers
27 views

Homework help on eigenvalues function minimization

so I actually have two separate questions which are homework bonuses for my numerical methods course. Unfortunately, because of the time of the semester, our TAs are not available so I don't have many ...
0
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1answer
406 views

Find the number of roots of a polynomial using Rouche's Theorem

Use Rouche's theorem to find the number of roots of the polynomial $z^5+3z^2+1$ in the anulus $1<|z|<2$. I am looking for a solution to this problem. My thoughts: This is a topic that ...
2
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1answer
26 views

Let $p$ be a prime number $\ge2$ and $u = \cos\left(\frac 2p\pi\right)+i\sin\left(\frac2p\pi\right) \in \mathbb C$. Prove that …

Let $p$ be a prime number $\ge2$ and $u = \cos\left(\frac 2p\pi\right)+i\sin\left(\frac2p\pi\right) \in \mathbb C$. Prove that $u$ is root of $f(x)=x^{p-1}+x^{p}+...+x+1$. I know that $f(x)$ is ...
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0answers
70 views

Show that the polynomial $x^8 -x^7 +x^2 -x +15$ has no real root. [duplicate]

Please check if my method is correct. Solution : Let $$f(x) = x^8 - x^7 + x^2 -x +15 $$ Now, let $g(x)= x^8 -x^7=x^7(x - 1)$ and $h(x)= x^2 -x=x(x-1)$. Thus, $$f(x)=g(x) + h(x) + 15$$ On analyzing $g(...
4
votes
5answers
475 views

Show that some of the root of the polynomial is not real.

\begin{equation*} p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_3x^3+x^2+x+1. \end{equation*} All the coefficients are real. Show that some of the roots are not real. I don't have any idea how to do this, I ...
0
votes
4answers
96 views

Find the roots of the polynomial? (Cardano's Method)

$y^3-\frac7{12}y-\frac7{216}$ This is part of Cardano's method, so I've gotten my first root to be: $y_1=\sqrt[3]{\frac7{432}+i\sqrt{\frac{49}{6912}}}+\sqrt[3]{\frac7{432}-i\sqrt{\frac{49}{6912}}}$ ...
0
votes
2answers
72 views

Showing that an equation has a root in an interval

Show that the equation $x^4 - 7x^3 + 1 = 0$ has a root in the interval $[0,1]$. How would I go about working this out in steps?
2
votes
1answer
77 views

Prove $p(x)>0$ for $x>b$

This is a question from a past paper which I have no solution to. Let $p(x)=x^n + a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}, n\geq 1$ be a polynomial of dgree n and let $b=1+|a_{1}|+\cdots +|a_{n-1}|+|a_{n}...
4
votes
1answer
56 views

$\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$

Question : Solve $\int\limits_{0}^{32/9}\sqrt{1+\frac{9x}{4}}dx$ My Try: Let u = $1+\frac{9x}{4}$ Then, $$du = \frac{9x}{4}dx$$ $$dx = \frac{4du}{9}$$ Substituting the above in the main equation,...
3
votes
2answers
89 views

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$

Prove that $f(x)=x$ can have at most one solution if $f'(x)\ne1$ What I did : Use $g(x) = f(x)-x$, then $g'(x) = f'(x)-1\ne0$ I suspect I have to use Rolle's theorem now, But I am having difficulty ...
14
votes
2answers
875 views

Is Vieta the only way out?

Let $a,b,c$ are the three roots of the equation $x^3-x-1=0$. Then find the equation whose roots are $\frac{1+a}{1-a}$,$\frac{1+b}{1-b}$,$\frac{1+c}{1-c}$. The only solution I could think of is by ...
1
vote
1answer
71 views

Location of the roots of $f'$ (Laguerre's theorem)

Let $f \in \mathbb{R}[X]$ be a polynomial of degree $n$ having $n$ distinct roots $a_1,...,a_n$. Let $b_1<...<b_{n-1}$ be the roots of its derivative $f'$ (note that $b_i \in ]a_{i}, a_{i+1}[$ ...
0
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0answers
41 views

Positive Zeroes within a Polynomial

Question: Let $a,b>0.$ Can the polynomial $$x^{10} − x^7 + 2x^5 + ax^3 − bx + 1$$ have exactly three (counting multiplicity) positive zeroes? Can it have three simple positive zeroes together with ...
2
votes
1answer
53 views

Confusion about exponents like ${x^m}^{(1/n)}$.

I've been reading this post. It says that $\sqrt[m]{x^n} = x^{n\frac 1m}=x^{\frac mn}=x$ if $m=n$. Let's take $x=-2$, and $m=n=2$. Now we have, $\sqrt[2]{(-2)^2}=\sqrt[2]{4}=2$ But according to that ...
0
votes
2answers
318 views

Extreme point of quadratic equation

For the below question read here: Write a function quadratic that returns the interval of all values $f(t)$ such that $t$ is in the argument interval $x$ and $f(t)$ is a quadratic function: $$...