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
44 views

If $\tan\alpha,\, \tan\beta,\,\tan\gamma$ are roots of $au^ 3 +(2a-x)u+ y=0$

If $\tan\alpha,\, \tan\beta,\,\tan\gamma$ are roots of $au^ 3 +(2a-x)u+ y=0$ for fixed $x$ and $y$ and $\tan\alpha + \tan\beta = h$ Find $ah^3 +(2a-x)h$. Options are A) $y$ B) $-y$ C) $2a- x$ D) ...
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0answers
42 views

Analytical solution to equation $ \arctan(x)-k \arctan(x/k)=c$

For the equation:$$\arctan(x)-k \arctan(x/k)=c$$ which is part of a gasdynamics function called Prandtl–Meyer function, it is not difficult to find the solution numerically, however, I'm wondering, ...
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1answer
71 views

Number of solutions to an equation $6\ln(x^2+1)-e^x = 0$

I need to determine how many solutions the equation $6\ln(x^2+1)-e^x = 0$ has. I wanted to find monotonic intervals of this function and check function's values at local extrema so I calculate ...
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0answers
61 views

Irrational roots of polynomials with integer coefficients

Consider an equation of the form $$ p(x) = 0, $$ where $p(x) = \sum_{i=0}^{n} a_i x^i$ and all $a_i \in \mathbb{Z}$. Is there an algorithm which can tell us that this equation has a root of the form ...
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3answers
102 views

How to find the range of zeroes in a polynomial

I am writing a java program that finds all of the zeroes of a polynomial by bisection. The first step, clearly, is to iterate through integers in a certain range looking for sign changes. I could ...
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0answers
64 views

How to get the proper fixed point iteration function?

When we find the approximated root of a function $f(x)$ in an interval $[a,b]$ from the fixed point iteration method, we derive a new function $g(x)$ which has a fixed point as a root of $f(x)$. Is ...
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1answer
35 views

Showing that at least one of the equations has two real roots

Let's suppose that $b_1, b_2, c_1, c_2$ are real numbers. We know that $b_1b_2=2(c_1+c_2)$. The task is to prove that at least one of the equations $x^2+b_1x+c_1=0$, $x^2+b_2x+c_2=0$ has two real ...
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1answer
47 views

How do we know that $\tan z$ or $\cos z$ don't have singularities off of the real axis?

Since $\sin z$ is bounded, $\tan z = \frac{\sin z}{ \cos z}$ has singularities when $\cos z = 0$. We know $\cos z = 0$ for $z = \frac{\pi}{2} + n\pi$ for $n \in \mathbb{Z}$, but could it not also be ...
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2answers
48 views

Show that $x+e^{-Bx^2}\mbox{cos}(x)$ has only one root over all reals ($B>0$).

Let $B>0$ and define for all $x\in \mathbb{R}$ the function $f(x)=x+e^{-Bx^2}\mbox{cos}(x)$. Prove that $f$ has exactly one root over $\mathbb{R}$. My original attempt was to show that the ...
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1answer
192 views

Can the general septic be solved by infinitely nested radicals?

I. Quintic. The general quintic can be reduced to the form, $$x^5=p+x\tag1$$ $$x = \sqrt[5]{p+x}$$ Hence by an iterative process, $$x =\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+\sqrt[5]{p+x\dots}}}}$$ ...
2
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3answers
419 views

condition for a cubic to have a repeated root

To write a condition for a cubic to have 2 real roots, can I equate the fuction to its derivative? I.e let $y=ax^3+bx^2+cx+d$ $\frac{dy}{dx}=3ax^2+2bx+c$ setting y and the derivative equal to 0 ...
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3answers
38 views

Find the value of $a$ from the equation

The roots of the equation: $ax^2-(5a+2)x+9a=0$ are equal. Find the value of $a$ given that $a>0$.
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2answers
151 views

Extracting factor from quadrinomial

As I'v learned about polynomials, I run into this quadrinomial: $$x^3+300x^2+30000x-953125 = 0$$ I've been studied how to factor this quadrinomial but didn't quite understand how it's done, here is ...
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5answers
109 views

How to find the cubic with roots: $k$, $k^{-1}$ and $1-k$?

This is the second part of a question that asks the same thing but for a quadratic, that part seemed to be fine. The next part asks you to show that: $$x^3-\frac{3}{2}x^2-\frac{3}{2}x+1=0 $$ is the ...
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0answers
27 views

Proving collinearity of the roots without finding the roots.

I was solving the polynomial $2z^3-(3-3i)z^2-(1+i)=0$ and found that they were in fact collinear! My question is, is there a way to prove that they are collinear without explicitly finding the roots? ...
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1answer
27 views

How can I show this ratio is >1 for intervals of x,y

I come here from a substantial application in statistics where I have reason to belive that the following ratio (function) is $$f(X,Y)=\frac{1}{(2XY^2-X^2Y^2+X^2-2X+1)^{\frac{1}{2}}} \ge 1$$ for ...
1
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1answer
33 views

N-th roots equation

I am facing the following equation and I do not have any idea about how to solve it. $\dfrac{(n^c-1)^a}{n^{ac}}$ = $\dfrac{1}{2}$. I am free to choose $c$ (any constant). $a$ on the other hand can be ...
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4answers
70 views

Roots of complex number

Solve the equation $$z^3-1=0$$ Show that the roots are represented in an Argand diagram by the vertices of an equilateral triangle. (EDITED: Thank you for your quick respond)
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3answers
185 views

explicit solution for transcendental equation

Does anyone knows whether there is an explicit, analytical solution for transcendental equations of the form $A x + B \tanh(C x) + \coth(x) = 0$, where $A, B$, and $C$ are positive real constants?
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1answer
35 views

About uniqueness of interest yield

I am not sure this belong to this site, in case I will post it elsewhere. Let $P$ be the price of a bond, let $C_k$ the promised cash flow in year $k$. Then we define the interest yield $y$ as the ...
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0answers
54 views

Find an equation in $x$ and $k$

Find an equation in $x$ and $k$ if, $$6u-8v+2=k^2$$ $$u^{2}=1+2v^{2}$$ $$v=2xy$$ $$u=x^2+2xy-y^2$$ Since we have 4 equations, we can eliminate 3 variables. But somehow, I'm not able to find an ...
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1answer
54 views

Zeros in the polynomial ring $\mathbb{R} [X,Y]$

I know that for $p(X) \in \mathbb{R} [X]$, $a$ is a zero of $p(X) \iff (X-a)|p(X)$. But what would the statement be for $p(X) \in \mathbb{R}[X,Y]$? This question comes from an example in my ...
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2answers
87 views

Prove: p-mq | f(m) where 'm' is any integer

How to prove that $p-mq \mid f(m)$ where $m$ is any integer, $f(x) = A_0 + A_1 x + A_2 x^2 + ... + A_{n-1} x^{n-1} + A_n x^n$, $f(x)∈ ℤ[x]$, $p/q$ is a zero for $f(x)$ and $p$ and $q$ are coprime ...
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3answers
97 views

Prove that the equation $x^{2}-x\sin(x)-\cos(x)=0$ has only one root in the closed interval $(0,\infty)$.

Here's the graph (http://www.wolframalpha.com/input/?i=%28x%5E2%29-xsenx-cosx%3D0). The part I'm having trouble with is proving that the root is unique. I can use the intermediate value theorem to ...
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3answers
67 views

Determining polynomial from roots of another polynomial

I am working on an exercize and I know how to more bruteforcely solve it through pure algebra in its simplest form, but it's such a massive mess to demonstrate so I would like to see if there is ...
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2answers
73 views

Solve this equation $(2x^2-3)^2=4(x-1)^2$?

This is the way I solved: What should I do next? Should I factorize or take a t that represents something?
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2answers
34 views

How to solve a twin root equation?

I thought to factorize a sqrt(x), but I can't find out anything. I thought to multiply both sides with themselves four times, but I'm not sure that works.
1
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1answer
49 views

Solving equation with complex roots.

I have the following question. My problem lies in (c). Question a) Find the three roots of the equation $(w+5)(w+8)(w+9)=360$. b) Let $z_0=\sqrt{-2+6i}$, where $i^2={-1}$. Show that the solutions ...
0
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4answers
33 views

Dividing a whole number by another fraction that includes a Root?

The question is simplifying $$\frac{9}{\frac{9\sqrt{97}}{97}}$$ The program has told me the answer found is $\sqrt{97}$, but I cannot figure out how this answer is found. I also do not have a ...
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2answers
51 views

Show Newton's method can go wrong with two roots

If $f:\mathbb{R} \to \mathbb{R}$ is differentiable with at least two roots, I wish to show that Newton's method will not converge for some $x_0$. I know that $f'(x)$ has a zero, say at $z$. It ...
2
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1answer
91 views

Find the Number of Zeros of $14z^{100}-5e^z$ in the Unit Disc. What are their Multiplicities?

This is an old qual problem. I consider the function defined by $f(z)=14z^{100}-5e^z$ and apply Rouche's Theorem. Let $g(z)=14z^{100}$. Then for $z$ on the boundary of the unit disc, $\vert ...
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1answer
33 views

Solving the equation $\sqrt[3]{x^2 + 15} = 2\sqrt[3]{x+1}$

In this equation $$ \sqrt[3]{x^2 + 15} = 2\sqrt[3]{x+1} $$ if I try to put in the third exponent both sides and I get an equation with the roots 7 and 1.Are these roots the same for the first ...
2
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1answer
49 views

Write a biquadratic equation that has as roots the numbers $2$ and $2\sqrt{2}$

I thought the answer would be: $$(x^2 - 4)(x^2 - 8) = 0$$ but it has $4$ roots the positive and negative values. Which is the correct answer?
0
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1answer
33 views

Newton's method with Exponents with base e

Use Newton's method to approximate the indicated root of the equation $e^x$=x the function $e^x$-x =0 ; i tried to find the root but it seems that this function has positive value for all numbers ...
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0answers
26 views

Problem with the Bisection method

I have a problem by using the bisection method. I have to get a route of 2xcos(2x)-sin(2x)=0 in the interval (3,4) However by the first estimation, I got a positive number when I put f(3.5) ...
0
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2answers
77 views

Does $x^3-e^x+\frac{13}{4}=0$ have an analytical solution?

I'm a little rusty with my math and have forgotten a lot of techniques for solving equations. I tried using $\ln$ to get rid of $e^x$ but then I end up with $\ln(x^3+\frac{13}{4})=x$ and I'm stuck. ...
0
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1answer
126 views

if $a,b,c$ are the roots of $x^3-px^2+qx-r=0$, find the value of $(a+b-c)(b+c-a)(c+a-b)$

If $a,b,c$ are the roots of $ x^3-px^2+qx-r=0$, find the value of $(a+b-c)(b+c-a)(c+a-b):$ A) $p^3 -8r$ B) $4pq-p^3$ C) $4pq-p^3-8r$ D) $4pq-8r$ Solution: $$a+b+c= p$$ ...
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1answer
61 views

Is it possible to find an analytical solution for “x” in this equation?

In my research I have come across the equation $$\prod_i^n \left( \frac{a_i}{x} \right)^\frac{b_i}{x} = \prod_i^n (1-d_i)^{(1-b_i)c}$$ Is it possible to obtain $x$ from this analytically, or do I ...
2
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0answers
15 views

Find roots of $\sin(a\,x)\sin(b\,y)-r\,\sin(b\,x)\sin(a\,y)$

Given $a,b,r$, I would like to find the roots of $f$ on $\mathbb{R}_+^2$: $$f(x,y)=\sin(a\,x)\sin(b\,y)-r\,\sin(b\,x)\sin(a\,y)$$ As you can see below, the roots of $f$ are curves (in red), ...
3
votes
3answers
181 views

Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.

If it is known that the equation $$ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $$ has a (real) root, prove the inequality $$ a^2 + b^2 \geq 8. $$ I am stuck on this problem, though, it is a very easy problem for ...
0
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1answer
17 views

How to build a function that is tangent to a sinoidal function

I am trying to design a function f defined, for every $x>0$, by $ax$ where $a$ is a constant value that I am searching for. Now, the problem is, for a fixed $c>0$, find $a$ such that the ...
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1answer
62 views

Finding all solutions to an equation in complex numbers!

Find all solutions to the equation, $$(\omega^2+1)^4=\omega$$ In complex numbers! I tried the substitution $\omega=z^4$ but wasn't helpful...and the equation becomes more complicated by this way!
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0answers
96 views

Can even degree Legendre polynomials have roots in common?

I'm wondering whether the Legendre polynomials $P_m(x)$ and $P_{m+2k}(x)$, with $m$ even and $k \in \mathbb{N}^+$, can have common roots. For $k=1$ it is straightforward to show that there are no ...
0
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3answers
73 views

if $n$ is natural odd number then the polynom : $P(x)=x^n+ax^2+b$ has at the most 3 different roots

I have this problem : if $n$ is natural odd number then the polynom : $P(x)=x^n+ax^2+b$ has at the most 3 different roots. $$P(x)=x^n+ax^2+b$$ $$P'(x)=nx^{n-1}+2ax$$ $$P''(x)=n(n-1)x^{n-2}+2a$$ I ...
2
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0answers
34 views

Good method for finding roots that *usually* fall within an interval?

I've been using Brent's method to find the roots of a monotonic, nonlinear, non-differentiable function. The roots often fall within a known interval, but Brent's method fails if they occasionally ...
-1
votes
3answers
101 views

Find a coefficient of quadratic polynomial, given the sum of its root.

The sum of the zeros of $f(x) = x^2 − 3kx − 14$ is $3$. Find $k$. How can I start this question?
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3answers
51 views

How do I find all possible complex roots of a polynomial with a degree of 4?

The problem: find all possible complex roots of $$P(x)=x^4 + 1$$ and write it down in a form of $a+ib$. Any hints on how I should start?
27
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10answers
5k views

Is There A Polynomial That Has Infinitely Many Roots?

Is there a polynomial function $P(x)$ with real coefficients that has an infinite number of roots? What about if $P(x)$ is the null polynomial, $P(x)=0$ for all x?
-1
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1answer
59 views

Roots of polynomial of degree $n$

Can we find the roots of a polynomial of degree $n > 3$ and if so how do we do it ? If we are not able to do so is there a proof to this? Using the remainder theorem to plug in numbers is not an ...
1
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0answers
43 views

Comparing zeros of two functions

How to show that $ x_f $, a single zero of (possibly downward-slopping) function $ f(\alpha, \beta, x) $ is greater than $ x_g $, a single zero of other (possibly downward-slopping, too) function $ ...