Tagged Questions

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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0
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2answers
12 views

Given that equation is a positive constant, equal roots, find value of k

I am having trouble solving this equation. It reads... Given that the equation $kx^2+12x+k = 0$, where $k$ is a positive constant, has equal roots, find the value of $k$. I am not sure where to ...
1
vote
1answer
25 views

Which solution is the right one??

If we want to solve the equation $sec^2(x)$ for finding the all roots(real and complex), we have two ways: 1-Direct solving for $sec^2(x)=0$ 2-Or by convert the above equation to polynomial series as ...
4
votes
3answers
238 views

Finding the all roots of a polynomial by using Newton-Raphson method.

Is there a general formulation for finding all roots of a polynomial, especially the complex ones, by using the Newton-Raphson Method?
2
votes
6answers
89 views

Show that function $4x^5+x^3+2x+1=0$ has exactly one real root.

I am struggling with problems like this, just trying to grasp the concept. Using the Intermediate Value Theorem and Rolle's Theorem, prove that $4x^5+x^3+2x+1=0$ has exactly one real root.
0
votes
1answer
24 views

How many roots does a complex polynomial has?

Define $f(z)=z^4-4z^3+8z-2$. Find how many zeros (including multiplicity) the function has in $\{z\in\mathbb{C}:|z|<3\}$. I tried using Rouché's-theorem on $\{z\in\mathbb{C}:|z|<3\}$. The ...
2
votes
0answers
14 views

Help me find $S_{-2}$ using the polynomial

I found the equation , then I found $S_3$ : $$S_3-S_1-3=0$$ And I found $S_3$ Also I gave it a try using the Generalized form: $$S_n+S_{n-2}+S_{n-3}=0$$ Let $n=0$ $$S_1+S_{-1}+S_{-2}=0$$ Now ...
0
votes
3answers
28 views

Roots and Cubic equations

Let $\alpha$, $\beta$ and $\gamma$ be the roots of the equation $2x^3 + 4x^2 + 3x - 1 = 0$. Calculate $\frac{1}{\alpha^2 \beta^2} + \frac{1}{\beta^2 \gamma^2} + \frac{1}{\alpha^2 \gamma^2}$ GIVEN ...
1
vote
2answers
36 views

An equation for the third powers of the roots of a given quadradic polynomial

The roots of the equation $3x^2-4x+1=0$ are $\alpha$ and $\beta$. Find the equation with integer coefficients that has roots $\alpha^3$ and $\beta^3$. GIVEN SR: $\alpha + \beta = \frac43$ PR: ...
3
votes
2answers
194 views

What method can i use to find the first 3 roots of y(t)=tan(t)+t?

Just by looking at the function: $$y(t) = \tan(t)+t$$ I can immediately see that there is a root at $t=0$, though after graphing it I can see many more roots and I can calculate them using computer ...
1
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0answers
22 views
3
votes
1answer
50 views

Riemann Zeta Function at $s = 1 + 2 \pi i n / \ln 2$

I am aware that the function defined by $$ \zeta(s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots $$ for $Re(s)>1$ can be extended to a function defined for $Re(s)>0$ by writing $$ ...
7
votes
1answer
51 views

Inequality relating coefficients and roots of a complex polynomial

While going through some olympiad handouts I stumbled upon a problem related to an upper bound for the Mahler measure, which stated that Given a polynomial $f(x) = x^n + a_{n-1}x^{n-1} + \dots + a_0 ...
0
votes
3answers
38 views

quadratics equation tricky problem

I am confused with this question- if $ax^2+bx+c$ have no real roots then- $1+c/a+b/a$ is-- a. Positive b. Negative c. Zero d. Can.t say I tried attempting it as follows $b^2-4ac<0$ so ...
7
votes
2answers
115 views

Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?
3
votes
4answers
151 views

Showing a function has one root in an interval

Could anyone shine some light on this question please? By considering $f'(x)$, show that $$f(x)=x^3 - 2$$ has exactly one root for $x$ greater than or equal to $0$.
2
votes
1answer
23 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 ...
5
votes
3answers
191 views

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
18 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 = ...
2
votes
3answers
45 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 $\frac{ln(x)}{a}$.
2
votes
4answers
39 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
265 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
108 views

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

Mathematica knows how to solve: ...
1
vote
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
110 views

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
40 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
38 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
44 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
223 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
46 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
74 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 ...
1
vote
2answers
171 views

$x^3+b^2x^2+2x+3=0$,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
63 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
89 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
vote
3answers
56 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
vote
2answers
46 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
57 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
72 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
28 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
44 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
467 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 ...