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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Root finding: Distance b/w 2 objects = 0. (and other examples of finding roots?)

Can someone explain general uses of finding roots? I understand you can find roots to help manually graph a function, but there's gotta be more. For example, in video games, I recall something about ...
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1answer
24 views

Existence of roots of $A_1\sin(\omega_1t+\phi_1)+A_2\sin(\omega_2t+\phi_2)$

It seems very intuitive that $$f(t)=A_1\sin(\omega_1t+\phi_1)+A_2\sin(\omega_2t+\phi_2)$$ has roots, but how to prove it? $A_i>0$, $\omega_i>0$ and $\phi_i\geqslant0$ (even though these ...
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0answers
26 views

Prove regarding number of distinct polynomial roots [duplicate]

Prove that for any g(x) in a $\mathbb{Z}/p\mathbb{Z}[x]$ field, the degree of $\gcd{(x^p-x,g(x))}$ is equal to the number of distinct roots of g(x) First of all, I am assuming that the "number of ...
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1answer
23 views

How to calculate $\alpha '\beta '+\alpha '\gamma ' + \beta '\gamma '$ when finding roots for a cubic equation?

Given the roots of the cubic equation $x^3+4x^2+3x+2=0$ are $\alpha, \beta, \gamma$, determine the cubic equation with roots $\beta\gamma, \gamma\alpha, \alpha\beta$. How on earth do I work out what ...
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1answer
46 views

Root Finding for Functions with many maxima and minima

I wondering if anyone can provide advice on the best combination of algorithms to find the roots (or any one root) of a function which is "dense" in that it has many local maxima and minima for ...
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1answer
183 views

Number of roots of a polynomial over a finite field

For any $g$ in $\mathbb{Z}/p\mathbb{Z}[x]$ prove that the degree of $f = \gcd(x^p - x, g(x))$ is exactly the number of distinct roots of $g$ in $\mathbb{Z}/p\mathbb{Z}$. My main problem is that I ...
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0answers
38 views

Uniqueness of $n$-th root in Hilbert space

Let $H%$ be a Hilbert space and $A \in \mathcal L(H)$, $A = A^*$, $A \geq 0$. Let $B = \sqrt[n]A$, where $n \geq 3$, i.e. $B \in \mathcal L(H)$, $B \geq 0$, $B^n = A$. How to show that such operator ...
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0answers
38 views

How to find the nature of roots of a given polynomial f(x) where nature of the coefficients is known?

Given the polynomial $ f(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x - a_0 $ where all coefficients are positive, what is the nature of the roots of the polynomial ?
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3answers
26 views

Nature of roots of a quadratic.

Question: If the equation $x^2+2x+1+\lambda=0$ has real and unequal roots, determine the nature of the roots of the equation $(\lambda+2)(x^2+2x+1+\lambda)=2\lambda(x^2+1)$. My attempt: Taking ...
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2answers
29 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 ...
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1answer
35 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 ...
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3answers
321 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?
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6answers
105 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.
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1answer
28 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 ...
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0answers
19 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 ...
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3answers
32 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 ...
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2answers
37 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: ...
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2answers
197 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 ...
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1answer
101 views

Solve $x^5 + x - 1 = 0$

Solve $x^5 +x - 1 = 0$ I am simply curios to see how the solution would go, since it is a quintic, it cannot be done by regular methods. Im just curios to see what people come up with (I can't solve ...
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1answer
56 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 $$ ...
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1answer
65 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 ...
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3answers
43 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 ...
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2answers
125 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?
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4answers
159 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
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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 ...
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3answers
193 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 ...
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1answer
19 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 = ...
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3answers
46 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 $\frac{ln(x)}{a}$.
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4answers
45 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 ...
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2answers
268 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 ...
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0answers
126 views

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

Mathematica knows how to solve: ...
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1answer
52 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) = ...
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1answer
39 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 $p$ ...
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2answers
26 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
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1answer
113 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. ...
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3answers
43 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) + ...
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1answer
42 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 ...
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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 ...
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3answers
230 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 + ...
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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 ...
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3answers
75 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 ...
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2answers
172 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. $$P(x)=x^3+b^2x^2+2x+3$$ ...
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1answer
78 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.
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1answer
67 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
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1answer
104 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 $x$ ...
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3answers
58 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 ...
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1answer
87 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 $x^9-15x^6-87x^3-125$ )
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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
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1answer
58 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) ...
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2answers
49 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 ...