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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2answers
40 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
35 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 ...
0
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0answers
18 views

About roots of multivariable complex polynomials.

We have a function $f : \mathbb{C}^2 \rightarrow \mathbb{C}$ such that, $f(z_1,z_2) = \prod_{i} (z_1 - a_i) = A(z_2-b)(z_2-c) $ where $a_i$ are known to be real. Now say $T$ is an operator which ...
0
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1answer
32 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
38 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
votes
1answer
26 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 ...
0
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0answers
18 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
67 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
votes
1answer
52 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: $$a+b+c= p$$ ...
2
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1answer
55 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
votes
0answers
14 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), ...
2
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3answers
84 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
votes
1answer
16 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 ...
1
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1answer
46 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!
4
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0answers
55 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 roots in common. For $k=1$ it is straightforward to prove this (see proof ...
0
votes
3answers
62 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
votes
0answers
25 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 ...
0
votes
3answers
56 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?
0
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3answers
44 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?
22
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9answers
3k 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?
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1answer
51 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
21 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 $ ...
3
votes
4answers
433 views

What does it mean to solve or find solutions in mathematics?

Something that has been really confusing me lately is that this equation has four solutions $$3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$$ But what does that mean? Until now solutions to me has meant, what are ...
0
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1answer
44 views

simple question on how to find complex roots

I have the following: What are the solutions to: $$e^{z^{2}}=-1$$ in the circle around $z=0$ of radius $R=2$ Apllying the complex $LOG$ I derived: $$z^2 = \pi*i(1+2k)$$ What is the simplest way ...
0
votes
2answers
15 views

The additive inverse for negative values only (otherwise zero)

I want to create a formula that only applies the additive inverse for negative values because I am trying to come up with a simple formula whereby two numbers are entered and if the second is larger ...
2
votes
4answers
81 views

find $z$ that satisfies $z^2=3+4i$

Super basic question but some reason either I'm not doing this right or something is wrong. The best route usually with these questions is to transform $3+4i$ to $re^{it}$ representation. Ok, so ...
2
votes
2answers
38 views

Is it possible to find an integer solution $r≥4$ to an equation?

Is it possible to find an integer solution $r≥4$ to this equation? $$11r²³-7r²¹+11r¹⁸-7r¹⁶-2r¹²+11r¹¹- 7r⁹-2r⁷-2 =0$$ I try some special values of $r$ but without any sucess.
0
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2answers
41 views

Is there any geometric interpretation or significance of the complex roots of a derivative?

I was doing some reading online when I stumbled here and learned about this geometric way of viewing the complex roots of a function. It got me thinking; the zeros of the derivative of a function $f$ ...
2
votes
1answer
56 views

How small would $|x_0 - a|$ be in order for $f(x)$ to converge to a for Newton's Method

I found that $f(x) = \cos(x) + \sin(50x)^2$ has a root $a = \pi/2$. Whenever we take our initial value $x_0$ close to a we get convergence, if we are far away from a we do not get convergence to our ...
1
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1answer
26 views

Proof of the proposition $V(S)=V(\langle S \rangle )$

In my lecture notes we have the following: Proposition: $$V(S)=V(\langle S \rangle )$$ Proof: $$\langle S \rangle=\left \{\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in R=K[x_1, x_2, \dots , ...
2
votes
4answers
48 views

Given that the equation, $(k-1)x^2-2(k-1)x-(3k+1)=0$ has real roots, show that $k^2-k≥0$

I can get to $k^2-k≥0$ but only when I make $b^2$ negative. The problem is why would I make $b^2$ negative other than the fact that $b$ is negative in the original equation? The problem with this is ...
1
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2answers
40 views

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.
2
votes
0answers
35 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 ...
1
vote
1answer
28 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 ...
4
votes
2answers
129 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 ...
1
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0answers
30 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) ...
4
votes
1answer
69 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 ...
1
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1answer
95 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 ...
1
vote
1answer
31 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 ...
1
vote
1answer
34 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
votes
2answers
49 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 ...
1
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1answer
48 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 ...
-1
votes
1answer
63 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
votes
1answer
55 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: ...
0
votes
0answers
30 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! $$a-b{{e}^{cx}}-{{e}^{\left( c+d \right)x}}=0$$
0
votes
0answers
46 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 ...
3
votes
4answers
106 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 $x \in \mathbb{C}$?
2
votes
3answers
85 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.
-1
votes
1answer
36 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
votes
1answer
23 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 ...