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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2
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1answer
16 views

How to find $α^2(β^4 +γ^4 +δ^4)+β^2(γ^4 +δ^4 +α^4)+γ^2(δ^4 +α^4 +β^4)+δ^2(α^4 +β^4 +γ^4)$

How to do the part (iv) . Please help. Here are my answers to the first parts: (i) α a root of given equation $\implies \alpha^4-5 \alpha^2 + 2 \alpha -1 = 0$ $\implies \alpha^{n+4} - 5 ...
1
vote
1answer
94 views
+50

Find the number of roots of the equation in $\mathbb{R}$

How many roots does the equation $$\\x^{x^x}=(x^x)^x\\$$ have in $\\\mathbb{R}$? My observations:I observed that $x=-1,1,2$ are its roots. Are there other roots of this equation?And how we can find ...
1
vote
1answer
17 views

Evaluating cubic roots of a quadratic

If $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^2 + 4x -5 = 0$, evaluate $\alpha^3 + \beta^3$.. I know that $$\alpha + \beta = \frac{-b}{a}$$ and $$\alpha \beta = ...
1
vote
3answers
41 views

exact roots of $e^{ax}-x=0$

How can I find the general solution to (not a numerical approximation) $e^{ax}-x=0$ as a function of $a$. I thought maybe something like $\frac{ln(x)}{a}$.
1
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4answers
33 views

Limit of implicit function

For $v>0$, let $f(v)$ be the smallest positive solution $x$ of $$\sqrt{\left(\frac{v}{x}\right)^2-1}=\tan x.$$ It can be confirmed graphically that $f(v)$ exists for all $v>0$. How can I show ...
6
votes
2answers
259 views

Methods for determining which roots of a polynomial are inside of the unit circle?

Let's say I have a polynomial such as $$p(x) = x^4 + bx^3 + cx^2 + bx + 1.$$ I strongly suspect that, for any parameters, there are always two roots inside the unit circle and two roots outside of ...
0
votes
0answers
92 views

Is solving the quintic the obstacle to solving the Riemann hypothesis?

Mathematica knows how to solve: ...
1
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1answer
51 views

Roots of product of two polynomials is the union of the roots of each polynomial

I'm trying to prove this lemma: The roots of $P(x)*Q(x)$ is the union of the roots of $P(x)$ and $Q(x)$ for all $x$. It's trivially true, which is why I find it hard to prove. Let $r(x) = ...
0
votes
1answer
38 views

Polynomial approximation

Say that you have $n+1$ points on the interval $[a,b]$, let's call them $\{x_0,\dots,x_n\}$. Take any two different $y_1, y_2$, points on $[a,b]$. My goal is to show that there exists a polynomial $p$ ...
0
votes
2answers
24 views

Completely factor a polynomial using the rational root theorem and synthetic division

I am currently seriously confused. My problem, as stated above, is about completely factoring a polynomial. My question is, once you get your possible factors, how do you then simplify it down? Ill ...
5
votes
1answer
98 views
+50

Can the measure of zeroes of a harmonic function be positive?

Let $u$ be a non-constant harmonic function of two variables defined, say, in the unit disk (or on the half plane for example). It is known that $u$ can vanish on some lines, as it discussed in here. ...
2
votes
3answers
38 views

Show that a polynomial $P(x)$ has $r$ as a double root if and only if $P'(r)=0$ and $P(r)=0$

Assuming that $r$ is a double root. Then $$P(x)=(x-r)^2\cdot k(x).$$ We also have the derivative: $$P'(x) = 2(x-r)k(x) + (x-r)^2k'(x).$$ Hence, $$P(r) = (r-r^2)k(r)=0$$ and $$P'(r) = 2(r-r)k(r) + ...
1
vote
1answer
34 views

polynomial over finite field, roots forming additive subgroup

Let $q=2^h$ and $t=2^r$ for some $h\ge r$ and denote by $\mathbb{F}_q$ the finite field of order $q$. (since the previous, simple version was wrong, I'm posting here a new version) Let $f$ be a ...
4
votes
0answers
42 views

How many iterations of the Newton's method are needed to achieve a given precision

There is a formula for bisection method to estimate number of iterations that are needed to achieve a given precision (desired significant figures) in the interval $[a,b]$ $$ n\ge ...
8
votes
3answers
220 views

Properties of Roots of polynomials

Today in highschool we were doing a chapter called "Roots of polynomials" where we learnt something new and interesting which is : $ax^2+bx+c=0$ Has roots $\alpha$ , $\beta$ Then: $$\alpha + ...
1
vote
2answers
44 views

If $\frac{[x_n]^n [1-[x_n]^n]}{(1-x_n) n} = a$, is $[x_n]^n$ increasing for $n\geq 3$?

Let $x_n$ be the solution to $\frac{x^n [1-x^n]}{(1-x) n} = a$, where $x \in [0,1], a \in [0,1]$ and $n \in \mathbf{N}$. I want to prove that $[x_n]^n$ is increasing in $n$ for $n\geq 3$. (From ...
1
vote
3answers
73 views

How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator?

Find all solutions of the equation $\tan(x) = 2 + \tan(3x)$ where $0<x<2\pi$. By replacing $\tan(3x)$ with $\dfrac{\tan(2x) + \tan(x)}{1-\tan(2x)\tan(x)}$ I've gotten to $\tan^3 (x) - 3 \tan^2 ...
0
votes
0answers
69 views
+50

$x^3+b^2x^2+2x+3=0$,Find several integer values of b such that the equation has roots.

$x^3+b^2x^2+2x+3=0$, Find several integer values of $b$ such that the equation has roots. My solution: I use the rational root theorem. $-3,3$ can be its rational roots. $$P(x)=x^3+b^2x^2+2x+3$$ ...
3
votes
0answers
61 views

How to solve this equation in radicals?

How to solve the equation $x^6-2\varphi^5x^5+2\varphi x+\varphi^6=0$ in radicals? where $\varphi$ is the golden ratio.
-2
votes
1answer
66 views

How to Solve $-3^x+617x+1625=0$

can anyone please help me solve this : $$-3^x+617x+1625=0$$ I can't do it analytically. originally the problem was to find intersection point of $$y=1625+617x$$ and $$y=3^x$$ i did the regular ...
3
votes
1answer
82 views

Cube root of complex number without trigonometric functions

Is there a general equation for a cube root of a complex number that does not exploit De Moivre's Theorem or in any way use trigonometric functions? For example, a square root of a complex number $x$ ...
1
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3answers
55 views

Roots of this third degree polynomial

I've got the following polynomial $$ x^3-6x^2-2x+40 $$ and I want to find its roots. The only option I see at the moment is to compute all the divisors of $40$ and their inverse, and manually check if ...
2
votes
1answer
85 views

How to find the polynomial which has the sum of two cube roots as one of its roots?

For example. How do I find the polynomial which has $\sqrt[3]2 + \sqrt[3]3$ as one of its roots? ( Hint: polynomial is $x^9-15x^6-87x^3-125$ )
0
votes
0answers
14 views

Find roots of $\sum_i \alpha_i\,\cos(\beta_i\,t)$

I'd like to solve analytically the following equation, where $\alpha_i$ and $\beta_i$ have known values in $\mathbb{R}$: \begin{equation} \sum_{i\leqslant N} \alpha_i\,\cos(\beta_i\,t)=0 ...
5
votes
1answer
55 views

Determinant of a matrice $a_{ij}=e^{a_ib_j}$

1) Let $a_1<\dots<a_n$ real numbers and $\lambda_1,\dots,\lambda_n\in\mathbb{R}\backslash\{0\}$ Let $f(x)=\lambda_1e^{a_1x}+\dots+\lambda_ne^{a_nx}$ Show that $f$ has at most $n-1$ zeroes 2) ...
1
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2answers
45 views

Finding more than one root using Newton's Method

The problem is stated as follows: Find the two roots of $x^{4}-8x^{2}-x+16 \:\:in \: [1,3].$ What is a good first guess / a good way to make a first guess when more than one root is involved, if one ...
3
votes
2answers
56 views

Zeros of a function of degree 4

I'm trying to show that the following function has no zeros $$ 60x^4-44x^3-25x^2-44x+60=0. $$ I already tried using Eisenstein's criterium, but since the first and the last coefficient are both $60$, ...
3
votes
0answers
31 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
0
votes
1answer
24 views

Solution of $p(z)=0$ with $z\in\mathbb C$ and $a_k\in\mathbb R$ for all $k$

Suppose $p(z)=a_0+...+a_nz^n$ with $a_k\in\mathbb R$ for all $k$. How can I prove that if $p(z)=0$ then $p(\bar z)=0$? I know it's true, but how can I prove it?
3
votes
5answers
70 views

How to solve $z^6+i=0$

I'm trying to solve $z^6+i=0$. I would have say that it's equivalent to $$z^6=-i\iff |z|^6e^{i6\arg(z)}=e^{i\frac{3\pi}{2}}\iff|z|^6=e^{i\left(\frac{3\pi}{2}-6\arg(z)\right)}$$ But I'm not able to ...
0
votes
0answers
26 views

Using Sturm sequences to locate the roots of a polynomial

So I've been doing the sequences and I understand the method of constructing a Sturm sequence but there is few things I don't get. Firstly, how does division using the remainder of division of fuction ...
0
votes
1answer
42 views

an exercise about finite extension field and root of a polynomial

Suppose $K|F$ is a field extension of degree $n$ and $f(X)\in F[X]$ is an irreducible polynomial of degree $m\ge 2$ and $(m,n)=1$. Prove that $f(X)$ has no root in $K$. thanks
1
vote
1answer
56 views

Confused by the solution of $x^3+bx^2+cx+d=0$

From $x^3 + bx^2 + cx + d = 0$, we have $(x-x_1)(x-x_2)(x-x_3)=0$ for some roots $x_1$, $x_2$ and $x_3$. Expanding this second expression gives us $$x^3 + \left(x_1+x_2+x_3\right)x^2 + \left(x_1x_2 + ...
1
vote
2answers
465 views

What does a complex root signify?

What does it tell me when I find that a polynomial has complex roots, except for the obvious fact that it crosses zero for these values? To me it seems that the existance of complex roots must have ...
0
votes
0answers
17 views

Roots of the Lagrange polynomials

This question follows my previous one Coefficients of Lagrange polynomials. Notations : $ n\in\mathbb{N}^*$ $[|1,n|]=\{1,2,\dots,n\}$ $A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers ...
1
vote
0answers
28 views

Location of Roots Symmetric Polynomial

I'm trying to prove (or disprove) that the roots of an even degree real symmetric coefficient polynomial are all on the unit circle. If it is not true, I will then try to find the conditions such that ...
0
votes
2answers
46 views

Roots of $x^{4} -28 x^{2}+49$ with Horner

I am studying Horner's algorithm and I got a problem I can't solve. The polynomal is $x^{4} -28 x^{2}+49$. After trying $\pm 1, \pm 7, \pm49$ with Horner I couldn't find any solution. Wolfram alpha ...
13
votes
1answer
195 views

Fitting a closed curve on the roots of ${x \choose k}-c$

Let $${n \choose k} = \frac1{(n+1) \operatorname{B}(n-k+1, k+1)}.$$ be the generalized binomial coefficient, and here $\operatorname{B}$ is the Beta function. Let $f_{k,c}(x)$ be the following ...
2
votes
0answers
46 views

Numerically solve for maximum root

I am looking for an efficient algorithm that can numerically solve a piecewise function for its maximum zero root. The piecewise function will normally take the form of the plots below where by below ...
0
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0answers
30 views

Show that the $n^{th}$ eigenfunction has $n-1$ zeros

Just a quick question. If I have an eigenfunction of a Sturm-Liouville form problem: $\phi_n = \sum_{n=1} A_n \sin\left[\frac{n\pi}{\log{b}}\log{x}\right]$, with $x$ between $1$ and $b$ - why is ...
0
votes
2answers
42 views

A Polynomial that Passes through the following four points?

I'm trying to do this for practice but I'm just going nowhere with it, I'd love to see some work and answers on it. Thanks :) Find a polynomial that passes through the points (-2,-1), (-1,7), ...
6
votes
0answers
67 views

Implications from $f(z)\in\mathbb{R} \Longleftrightarrow z\in \mathbb{R}$ [duplicate]

Let $f:D(0,1)\longrightarrow \mathbb{C}$ be a holomorphic function such that $f(z)\in\mathbb{R} \Longleftrightarrow z\in \mathbb{R}$. How to prove that $f$ has at most one zero on the disk. By ...
3
votes
1answer
28 views

Solving equations: reasoning doesn't work backwards?

In doing my (high school) math homework, I came to an issue that doesn't make sense to me. Given an equation $0 = a_1 + a_2x + a_3x^2 + \dots$, we can multiply both sides by $x$ to obtain $0 = a_1x + ...
2
votes
1answer
39 views

A positive polynomial is the sum of two squares in $\mathbb{R}[X]$ [duplicate]

Let $P\in\mathbb{R}[X]$ be a positive polynomial. I want to show that there exists $A,B\in\mathbb{R}[X]$ so that $P=A^2+B^2$ $\displaystyle ...
3
votes
2answers
82 views

Prove that $\prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1$

Prove that $$ \prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1. $$ In my some problem I have used $$ \prod_{k=0}^{7}(z-\mathrm{e}^{2k\pi i/8})=z^8-1. $$ I have verified this. So I think in general ...
0
votes
2answers
29 views

Stuck finding the zeros of a polynomial (complex and real)

Stuck finding the zeros of this polynomial (complex and real): $$x^4+2x^2+1$$ I am not sure how I would factor this. The constant value is really throwing me off. I just need a hint on how to get ...
1
vote
1answer
16 views

Horizontal Cylinder Gas Problem

We have a perfect cylinder with a diameter of 3 ft that lies horizontal. The gas gauge is broken so we are forced to use a dipstick to determine how much gas in our tank. In this problem we are ...
1
vote
1answer
18 views

Root of function involving trig and exponential

Would anyone know an analytical solution to finding the root of $$ f(x) = \sin(x^2) - e^x $$ in $[-1,1]$? I'm writing a simple root finding program and thought I'd try this as a test case, but ...
4
votes
0answers
64 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
0
votes
0answers
40 views

derivative and roots of polynomials

Given a polynomial $g(x)=\frac{f(x)}{(x-x_1)(x-x_2)$ can it be proven that the roots of $g'(x)=0$ would lie in the interval [x_1,x_2]? Real/Complex, im not sure