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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Given some of the roots of the function $f(x) = x^3+bx^2+cx+d$, how do I find the coefficients of that function?

Two of the roots of $f(x) = x^3+bx^2+cx+d$ are $3$ and $2+i$. How do I find b+c+d? The answer choices are -7, -5, 6, 9, and 25.
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0answers
30 views

Possible integer roots of polynomial with real coefficents

If $p\in\mathbb{Q}[X]$, then the rational root theorem gives us possible integer roots of $p$. If $p\in\mathbb{R}[X]$, the theorem cannot be applied. Nevertheless, triangular inequality gives us lower ...
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1answer
26 views

$K-$rational solution of the equation - Is $\mathbb{Q} \leq \mathbb{Q}_p$?

Let $P(x, y) \in \mathbb{Q}[x, y]$. We consider the equation $P(x, y)=0$. If $a, b \in \mathbb{Q}$ such that $P(a, b)=0$ then $(a, b) \in \mathbb{Q}^2$, is called a rational solution. If $K$ a ...
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2answers
118 views

Real solutions of $x^n + y^n = (x+y)^n$

I have to find all real solutions of the following equation: $x^n + y^n = (x+y)^n$ Clearly for $n = 1$, the equation holds for every $x,y$ real numbers. If $n$ is greater or equal to $2$, we do ...
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0answers
21 views

Root with bolzano theorem

Given this equation $a\cos{x}+b=x$ with $a,b>0$ how to prove that there is at least one root between $(0,a+b]$ ? For $x=0$ its $a+b$ which is >0 For $x=a+b$ its $a\cos(a+b) ...
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1answer
67 views

The diophantine equation $y^2=x^3+7$ has no solutions.

In my lecture notes there is the following example: The diophantine equation $y^2=x^3+7$ has no solutions. Proof: If the equation would have a solution, let $(x_0, y_0)$, $y_0^2=x_0^3+7$, then ...
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1answer
32 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
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0answers
26 views

Finding roots of a discrete complex valued function [on hold]

I am struggling with a numerical problem. I have a discrete dataset with complex valued numbers which are the function of a real variable. The function is a black box. Is there any way to find the ...
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1answer
89 views

Number of real roots of $2 \cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}$

Find the number of real roots of $ \cos \,\left(\dfrac{x^2+x}{6}\right)= \dfrac{2^x+2^{-x}}{2}$ 1) 0 2) 1 3) 2 4) None of these My guess is to approach it in graphical way. But equation seems ...
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1answer
28 views

Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
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4answers
156 views

How to solve current exponential equation? [on hold]

There is an equation: $$3^x + 7^x = 21^x$$ How to solve this?
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0answers
25 views

What is a one-parameter Newton's method?

The Newton's method that I know is defined as follows: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ However, I've recently encountered a paper that talks about a one-parameter family of Newton's method ...
2
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1answer
42 views

Inverting complex cosine

I have been working out problem 3a in chapter 1 section 3 in Basic Complex Analysis by Marsden. He asks to solve $$ \cos z=\frac{3}{4}+\frac{i}{4} $$ After putting cosine in its exponential form and ...
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1answer
45 views

Roots less than 1 if at least one coefficient is greater than one

I have this doubt. If you have this equation with $\alpha_i \in \mathbb R$ $$P(z)=1-\alpha_{1}z-\alpha_{2}z^{2}- \cdots - \alpha_{p}z^{p}=0$$ I believe that if there exist an $\alpha$ greater or equal ...
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1answer
54 views

Closed form of $\cot x=x$

I plotted the graphs of $y=\cot x$ and $y=x$. Its clear that they have infinite intersections. I tried to solve for the first root but it doesn't seem to be any known number to me. Even Wolfram Alpha ...
2
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1answer
18 views

Sixth root of -64 using Euler's formula and De Moivre's theorem

I am attempting to solve: $$(-64)^{\frac{1}{6}}$$ Using the relation: $$a+bi=re^{i(\tan^{-1}(\frac{b}{a})+2\pi n)}$$ And then applying De Moivre's theorem: ...
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0answers
28 views

root of $a-b{{e}^{cx}}-{{e}^{\left( c+d \right)x}}=0$

I am trying to find the root(s) of this equation, basically write variable x in terms of parameters a, b, c, and d. not sure how to proceed. Thanks! $$a-b{{e}^{cx}}-{{e}^{\left( c+d \right)x}}=0$$
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0answers
34 views

Different ways to prove Fundamental Theorem of Algebra

This is just a curosity .I know some proofs of the fact that Every non constant polynomial with complex coefficient has a complex root via using Liouville's theorem in Complex Analysis.Proof goes as ...
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4answers
96 views

Is the zero polynomial the only polynomial that vanishes at every point of $\mathbb C$?

The zero polynomial has the property that every value it takes on $\mathbb C$ is zero. Is the converse true, or are there other polynomials $f$ such that $ f(x)=0$, for all $x \in \mathbb{C}$?
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3answers
83 views

How many $n$th roots does $0$ have?

Do we say that $0$ has $n$ $n$th roots, all nondistinct, or only one? I don't think it makes any difference, but I'm curious what the convention is.
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1answer
35 views

Find the algebraic set $V(S)$

How can we find the algebraic set $$V(x^2+y^2-1)$$ ? $$V(S)=\{(a_1, a_2, \dots , a_n ) \in K^n |f_a(a_1, a_2, \dots , a_n )=0, \forall a \in A\}$$ where $$S=\{f_a \in K[x_1, x_2 , \dots , x_n] | a \in ...
2
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1answer
18 views

Find the maximum number of a continuous function

Lets define a function $z:\mathbb{R}^\mathbb{R}\to\mathcal P(\mathbb R)$ that gives you the set of zeros of any $\mathbb R ^\mathbb R$ function. Now, we define a set $S=\{z(f):f\in\mathbb R ^\mathbb ...
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2answers
29 views

Problem with the rational root theorem

Consider this polynomial: $f(x)=(2x+5)(x-3)(x+8/3)=0$. Then $f(x)=2x^3+...+(-40)$ Here is a list of all factors of $40$ and $2$: $40$: $±1$, $±2$, $±4$, $±5$, $±8$, $±10$, $±20$ $2$: $±2$, $±1$ ...
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1answer
26 views

Fourth Order Homogeneous Ordinary Differential Equation With Double Complex Conjugate Roots (2.10-14)

This is actually a problem in algebra as shall be seen. I need to find the general solution for the following differential equation: $$y''''+8y''+16y=0$$ The characteristic equation for this is: ...
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1answer
43 views

Use Newton's method to find root for the following equations

I have to use Newton's method to find the roots with accuracy $10^{-5}$ of the following equation : $e^{x} + 2^{-x} +2\cos x -6 =0$ in the interval $(1,2)$ So $f'(x)= e^x - [2^{-x}]*[\log(2)] ...
1
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1answer
34 views

Roots of polynomial equation $2a x^\gamma + ax^{\gamma - 1} - 2 = 0$

I would like to find roots of the following polynomial equation $$2a x^\gamma + ax^{\gamma - 1} - 2 = 0$$ where $a,\gamma>0$ (we might also assume that $\gamma \in\mathbb{N}$ if needed). Playing a ...
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1answer
48 views

Finding roots of a complex polynomial in $\{\operatorname{Re}(z) < 0 \} $

How many roots does $P(z)=2z^4+z^3-5z^2+z+2$ have in $\{ \operatorname{Re}(z)<0 \}$? I was told that I should compute $P(it)$ for $t \in \Bbb R$ which is: $P(it)=2t^4 +5t^2 +2 +it(1-t^2) $. ...
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2answers
60 views

If square root is the inverse function of $5^2$ what is the inverse function of $5^1$

I am not great at maths or anything, but just had a general question: If square root is the opposite of $5^2$, what is the opposite of $5^1$, $5^3$, $5^4$? Is there an opposite? How would I work it ...
3
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1answer
85 views

Geometry: How to find cube root, fourth root, fifth root… and so on?

As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$. What I want to know is how to find cube root, fourth root, fifth root or ...
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1answer
21 views

Indefinite integral fractional roots

I know the result of these indefinite integrals, but I don´t understand how the calculaton gets there: $$\int \frac{1}{\sqrt{x}}dx = 2 \sqrt{x}$$ $$\int \frac{1}{\sqrt[3]{x}}dx = ...
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1answer
32 views

Degree of min distance function between two algebraic curves

Suppose I have two algebraic curves $C_1$ and $C_2$ in the plane. I would like to find the minimum distance between the two curves. If the two curves have degrees $n_1$ and $n_2$, what is ...
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3answers
80 views

Is it possible to find the complex roots of $x^3 + 2 x^2 - 3 = 0$

Sorry to ask so many roots questions in such short bursts, but I want to know if it is humanly possible to compute the complex root of $$x^3 + 2 x^2 - 3 = 0$$ through algebraic manipulation? ...
0
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5answers
80 views

How to compute the roots of $x^3 - x^2 - 4x + 4$ = $0$

I am wondering whether there is a simple way to find the roots of $x^3 - x^2 - 4x + 4$ = $0$ by algebraic manipulation I will accept if this is not a trivial equation to compute the roots of Any ...
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0answers
23 views

Integer roots of polynomial

I have a polynomial with integer coefficients on the form $$ p_{l,N}(x) = x^{2l+1} - 2\sum_{m=0}^{l}\binom{2l+1}{2m}x^{2m}\sum_{j=1}^N j^{2l+1-2m} $$ and I am interested in finding the possible ...
2
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0answers
36 views

Number of integral solutions to a polynomial

Given a polynomial of $n$th order, represented by $$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+a_{n-1}x+a_{n}=0$$ Is it possible to find the number of integral solutions/roots to ...
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1answer
34 views

reconstructing a quadratic equation from roots

I have this quadratic equation $$2x^2+x-3=0$$ that I wish to reconstruct from its roots. $$D=b^2-4ac=25$$ $$x_1=\frac {-b\pm \sqrt D} {2a} = 1 \text{ and } \frac {-2} 3$$ Now, I've always learned that ...
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1answer
37 views

How to get the polynomial which roots are almost “equal and opposite of sign ?”

Respected All I got stuck in it and need your help. We know that if $\alpha_1, \cdots, \alpha_5$ be the roots of $p(x):=x^5+ax^4+bx^3+cx^2+dx+e=0$ then the equation which roots are opposite in sign ...
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1answer
44 views

find a quadratic polynomial p ( x ) and a number n such that p ( x ) and a number $n \pmod n $ has at least 2015 roots?

I understand what the question is asking for, but I don't know how to prove my answer. Let's say I took an equation of the form: $x^2+ 6x+ 8 \equiv0 \pmod {15}$. The first four roots are ...
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1answer
43 views

Secant method and false position method exercise

We have $f(x)=x^2-6$. I have to find $p_3$ if $p_0 = 3$ and $p_1 = 2$ by using a) Secant method b) False position method So for the first one I have $p_2=p_0- \dfrac ...
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2answers
41 views

Nature of The Roots of The Quadratic Equation $(a-1)x^2+(4a-2)x+4a+1=0$ [closed]

For which values of the real parameter $a$ are the roots of the quadratic equation: $$ (a-1)x^2+(4a-2)x+4a+1=0 $$ a) Real b) Positive
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1answer
22 views

Slight problem with solving a trigonometric equation.

I had to prove the identity $4\cos^3x-3\cos x=\cos 3x$ and then use it to solve the equation $(4\cos^2x-3)(4\cos^23x-3)(4\cos^29x-3)=1$. After proving the identity I proceeded to simplify the ...
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2answers
36 views

Find root of equation using bisection method?

Question : Find an approximate value of $\sqrt[3]{25}$ using Bisection Method. Since it doesnt state the accuracy in the question,how many iterations am I going to do to get that approximate value? ...
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1answer
50 views

What is meant by positive root of $x^3-x^3-1$?

I am a bit confused. I think there must be a mistake. In a text I read: The entropy is $2\ln p$, where ...
1
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4answers
70 views

how to factor this cubic polynomial

Let $f(t)=36t^3-19t+5$ be a cubic polynomial. How we can factor $f$ to its roots? Mathematica says that $f(t)=(-1+2 t) (-1+3 t) (5+6 t)$. How?
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1answer
58 views

Multivariate polynomials at bounded evens

Univariate polynomials Given $n$, is there a degree $cn^{c'}$ polynomial $p(x)\in\Bbb R[x]$ and a degree $dn^{d'}$ polynomial $q(x)\in\Bbb R[x]$ with fixed $c,c',d,d'>0$ such that $$m\in\Bbb ...
4
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0answers
137 views

Prove equation has only one root in a specific interval

Prove that the following equation has only one solution in the interval $[-\text{min}(a_i), +\infty]$: $f(x) = \left(\sum_{i=1}^n \frac{1}{a_i + x}\right)\times \left(\sum_{i=1}^n \frac{a_i b_i}{(a_i ...
6
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2answers
101 views

Degree Polynomials and Zeroes

"Find a degree $3$ polynomial that has zeros $-3, 4$ and $8$ and in which the coefficient of $x^2$ is $-18$." I've been trying to solve this problem, but I keep getting it wrong. I've worked with ...
5
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0answers
44 views

Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
1
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3answers
66 views

Polynomials in one variable with infinitely many roots.

Can a non-zero polynomial in one variable have infinitely many roots ? Can a non-zero polynomial in one variable have uncountably many roots ? Motivation : over $\mathbb Z/12\mathbb Z$, ...
4
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2answers
93 views

Find the number of polynomial zeros of $z^4-7z^3-2z^2+z-3=0$.

Find the number of solutions of $$z^4-7z^3-2z^2+z-3=0$$ inside the unit disc. The Rouche theorem fails obviously. Is there any other method that can help? I have known the answer by Matlab, but ...