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.

learn more… | top users | synonyms (1)

1
vote
1answer
48 views

Finding the argument of a complex number,

I'm trying to locate my four zeroes of a complex-valued function, in order to apply the Residue Theorem. After using the quadratic formula, I am left with $$z^2 = [-3 \pm i\sqrt7] / 2$$ writing the ...
2
votes
3answers
77 views

By using the Fixed-Point Iteration, I have to find the roots of $f(x)=x^2-x-1=0$

By using the Fixed-Point Iteration, I have to find the roots of $f(x)=x^2-x-1=0$ First I write it in terms $x=f(x)$ $$x^2=x+1$$ $$x=1+\frac{1}{x}$$ Then I make a sequence ...
2
votes
2answers
211 views

how to find the number of real root

Prove $\;ax^3+bx+c=0\;$, with $\;a,b>0\;$, has at most one real root. Should I use Rolle's theory for this? But I use it, what is the boundary of this function? I can only work out x must not be ...
0
votes
1answer
147 views

Steffensen's method in Numerical Analysis

In some sources, Steffensen's method is the development of Newton's method to avoid computing the derivative, http://bit.do/Um6M http://cims.nyu.edu/~donev/Teaching/NMI-Fall2010/Homework4.pdf ...
2
votes
1answer
53 views

How to find initial estimate of roots from graphs?

I have f1(x,y) = x^2 + 3y^2 - 1 = 0 and f2(x,y) = (x-2)^2 + (y-1)^2 - 4 = 0 I am suppossed to find the roots of these nonlinear ...
0
votes
0answers
26 views

Efficient method to calculate passes (rises and sets) for satellites

There is a function describing the characterisic elevation of ISS seen from an observers horizon. Calculating of an elevation at one time is pretty expensive. So I wanna try to avoid naive iterating ...
2
votes
6answers
332 views

Disproving existence of real root in some interval for a quintic equation

Disprove the statement: There is a real root of equation $\frac{1}{5}x^5+\frac{2}{3}x^3+2x=0$ on the interval (1,2). I am not sure whether to prove by counter-example or by assuming the statement is ...
1
vote
1answer
154 views

What is the condition for the first root of a cubic function to be positive?

Is there any way to determine if the "first root" of a cubic equation is positive, assuming that it's real, given coefficients $a,b,c,$ and $d$? I tried following along with Wikipedia's explaination ...
5
votes
1answer
177 views

$15a+6b+4c+8d=0$ implies $ax^3+bx^2+cx+d$ has a positive root

Let $a,b,c,d$ be real numbers such that $15a+6b+4c+8d=0$. Show that $f(x)=ax^3+bx^2+cx+d$ has a positive root. (Komal, Problem N. 170.) I want to try to use the intermediate value theorem, showing ...
2
votes
2answers
616 views

Finding the roots of $\tanh(x)-\tan(x)=0$

I solved this equation by Wolfram Alpha. How can I solve it analytically to find all roots $$\tanh(x)-\tan(x)=0$$
1
vote
0answers
56 views

Zeroes of polynomials with several variables

Just consider a polynomial with multiple variables on the field of real number, A is its zeroes set. Is A a zero measure set ? How to prove that?
1
vote
5answers
182 views

Number of real roots of the equation $(6-x)^4+(8-x)^4 = 16$

Find number of real roots of the equation $(6-x)^4+(8-x)^4 = 16$ $\bf{My\; Try::}$ Let $f(x) = (6-x)^4+(8-x)^4\;,$ and we have to find real values of $x$ for which $f(x) = 16$. Now we will form ...
2
votes
0answers
101 views

Lower bound on a polynomial far from its zeros

Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far ...
0
votes
0answers
82 views

Roots of polynomials: Vieta's Formula

Let $p_n(t) = c_0 + c_1 t + c_2 t^2 + \ldots + c_n t^n$ with $c_i \in \mathbb{Q}$ and let the roots of $p_n(t) = 0$ be $R = \{r_1, r_2, \ldots r_n \}$. Vieta's formula states that $\sum_{i=1}^n r_i = ...
2
votes
1answer
33 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 ...
2
votes
0answers
28 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 ...
0
votes
1answer
117 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 ...
3
votes
1answer
196 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 ...
1
vote
1answer
215 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 ...
0
votes
0answers
43 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 ...
0
votes
3answers
40 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 ...
1
vote
2answers
947 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 ...
0
votes
1answer
39 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 ...
8
votes
3answers
2k 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
140 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
57 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
21 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
53 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
47 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
208 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 ...
5
votes
1answer
129 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 ...
3
votes
1answer
114 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
126 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
68 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 ...
10
votes
3answers
241 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
247 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
28 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 ...
2
votes
2answers
70 views

Using descartes rule of sign

Use Descartes' rules of signs to discuss the possibilities for the roots of each equation. Do not solve equation. $$p(x)= x^3+5x^2+7x+1=0$$ $p(x)$ I saw no sign change $p(-x)$ I saw 2 sign ...
5
votes
3answers
203 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 ...
2
votes
1answer
38 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
64 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
273 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
530 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
175 views

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

Mathematica knows how to solve: ...
1
vote
1answer
80 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
62 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
63 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
138 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
100 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
69 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 ...