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
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 ...
1
vote
2answers
32 views

Problem with the rational root theorem

Consider this polynomial: $f(x)=(2x+5)(x-3)(x+8/3)=0$. Then $f(x)=2x^3+...+(-40)$ Here is a list of all factors of $40$ and $2$: $40$: $±1$, $±2$, $±4$, $±5$, $±8$, $±10$, $±20$ $2$: $±2$, $±1$ ...
1
vote
1answer
175 views

Fourth Order Homogeneous Ordinary Differential Equation With Double Complex Conjugate Roots (2.10-14)

This is actually a problem in algebra as shall be seen. I need to find the general solution for the following differential equation: $$y''''+8y''+16y=0$$ The characteristic equation for this is: ...
1
vote
1answer
50 views

Use Newton's method to find root for the following equations

I have to use Newton's method to find the roots with accuracy $10^{-5}$ of the following equation : $e^{x} + 2^{-x} +2\cos x -6 =0$ in the interval $(1,2)$ So $f'(x)= e^x - [2^{-x}]*[\log(2)] ...
1
vote
1answer
35 views

Roots of polynomial equation $2a x^\gamma + ax^{\gamma - 1} - 2 = 0$

I would like to find roots of the following polynomial equation $$2a x^\gamma + ax^{\gamma - 1} - 2 = 0$$ where $a,\gamma>0$ (we might also assume that $\gamma \in\mathbb{N}$ if needed). Playing a ...
1
vote
1answer
50 views

Finding roots of a complex polynomial in $\{\operatorname{Re}(z) < 0 \} $

How many roots does $P(z)=2z^4+z^3-5z^2+z+2$ have in $\{ \operatorname{Re}(z)<0 \}$? I was told that I should compute $P(it)$ for $t \in \Bbb R$ which is: $P(it)=2t^4 +5t^2 +2 +it(1-t^2) $. ...
0
votes
2answers
65 views

If square root is the inverse function of $5^2$ what is the inverse function of $5^1$

I am not great at maths or anything, but just had a general question: If square root is the opposite of $5^2$, what is the opposite of $5^1$, $5^3$, $5^4$? Is there an opposite? How would I work it ...
4
votes
1answer
166 views

Geometry: How to find cube root, fourth root, fifth root… and so on?

As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$. What I want to know is how to find cube root, fourth root, fifth root or ...
0
votes
1answer
25 views

Indefinite integral fractional roots

I know the result of these indefinite integrals, but I don´t understand how the calculaton gets there: $$\int \frac{1}{\sqrt{x}}dx = 2 \sqrt{x}$$ $$\int \frac{1}{\sqrt[3]{x}}dx = ...
1
vote
1answer
43 views

Degree of min distance function between two algebraic curves

Suppose I have two algebraic curves $C_1$ and $C_2$ in the plane. I would like to find the minimum distance between the two curves. If the two curves have degrees $n_1$ and $n_2$, what is ...
0
votes
3answers
83 views

Is it possible to find the complex roots of $x^3 + 2 x^2 - 3 = 0$

Sorry to ask so many roots questions in such short bursts, but I want to know if it is humanly possible to compute the complex root of $$x^3 + 2 x^2 - 3 = 0$$ through algebraic manipulation? ...
0
votes
5answers
82 views

How to compute the roots of $x^3 - x^2 - 4x + 4$ = $0$

I am wondering whether there is a simple way to find the roots of $x^3 - x^2 - 4x + 4$ = $0$ by algebraic manipulation I will accept if this is not a trivial equation to compute the roots of Any ...
1
vote
0answers
34 views

Integer roots of polynomial

I have a polynomial with integer coefficients on the form $$ p_{l,N}(x) = x^{2l+1} - 2\sum_{m=0}^{l}\binom{2l+1}{2m}x^{2m}\sum_{j=1}^N j^{2l+1-2m} $$ and I am interested in finding the possible ...
3
votes
0answers
49 views

Number of integral solutions to a polynomial

Given a polynomial of $n$th order, represented by $$f(x)=a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+a_{n-1}x+a_{n}=0$$ Is it possible to find the number of integral solutions/roots to ...
1
vote
1answer
51 views

reconstructing a quadratic equation from roots

I have this quadratic equation $$2x^2+x-3=0$$ that I wish to reconstruct from its roots. $$D=b^2-4ac=25$$ $$x_1=\frac {-b\pm \sqrt D} {2a} = 1 \text{ and } \frac {-2} 3$$ Now, I've always learned that ...
1
vote
1answer
52 views

How to get the polynomial which roots are almost “equal and opposite of sign ?”

Respected All I got stuck in it and need your help. We know that if $\alpha_1, \cdots, \alpha_5$ be the roots of $p(x):=x^5+ax^4+bx^3+cx^2+dx+e=0$ then the equation which roots are opposite in sign ...
1
vote
1answer
69 views

find a quadratic polynomial p ( x ) and a number n such that p ( x ) and a number $n \pmod n $ has at least 2015 roots?

I understand what the question is asking for, but I don't know how to prove my answer. Let's say I took an equation of the form: $x^2+ 6x+ 8 \equiv0 \pmod {15}$. The first four roots are ...
1
vote
1answer
92 views

Secant method and false position method exercise

We have $f(x)=x^2-6$. I have to find $p_3$ if $p_0 = 3$ and $p_1 = 2$ by using a) Secant method b) False position method So for the first one I have $p_2=p_0- \dfrac ...
-2
votes
2answers
53 views

Nature of The Roots of The Quadratic Equation $(a-1)x^2+(4a-2)x+4a+1=0$ [closed]

For which values of the real parameter $a$ are the roots of the quadratic equation: $$ (a-1)x^2+(4a-2)x+4a+1=0 $$ a) Real b) Positive
1
vote
1answer
26 views

Slight problem with solving a trigonometric equation.

I had to prove the identity $4\cos^3x-3\cos x=\cos 3x$ and then use it to solve the equation $(4\cos^2x-3)(4\cos^23x-3)(4\cos^29x-3)=1$. After proving the identity I proceeded to simplify the ...
1
vote
2answers
70 views

Find root of equation using bisection method?

Question : Find an approximate value of $\sqrt[3]{25}$ using Bisection Method. Since it doesnt state the accuracy in the question,how many iterations am I going to do to get that approximate value? ...
1
vote
1answer
54 views

What is meant by positive root of $x^3-x^3-1$?

I am a bit confused. I think there must be a mistake. In a text I read: The entropy is $2\ln p$, where ...
1
vote
4answers
75 views

how to factor this cubic polynomial

Let $f(t)=36t^3-19t+5$ be a cubic polynomial. How we can factor $f$ to its roots? Mathematica says that $f(t)=(-1+2 t) (-1+3 t) (5+6 t)$. How?
1
vote
1answer
58 views

Multivariate polynomials at bounded evens

Univariate polynomials Given $n$, is there a degree $cn^{c'}$ polynomial $p(x)\in\Bbb R[x]$ and a degree $dn^{d'}$ polynomial $q(x)\in\Bbb R[x]$ with fixed $c,c',d,d'>0$ such that $$m\in\Bbb ...
4
votes
0answers
199 views

Prove equation has only one root in a specific interval

Prove that the following equation has only one solution in the interval $[-\text{min}(a_i), +\infty]$: $f(x) = \left(\sum_{i=1}^n \frac{1}{a_i + x}\right)\times \left(\sum_{i=1}^n \frac{a_i b_i}{(a_i ...
6
votes
2answers
104 views

Degree Polynomials and Zeroes

"Find a degree $3$ polynomial that has zeros $-3, 4$ and $8$ and in which the coefficient of $x^2$ is $-18$." I've been trying to solve this problem, but I keep getting it wrong. I've worked with ...
5
votes
0answers
50 views

Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
1
vote
3answers
129 views

Polynomials in one variable with infinitely many roots.

Can a non-zero polynomial in one variable have infinitely many roots ? Can a non-zero polynomial in one variable have uncountably many roots ? Motivation : over $\mathbb Z/12\mathbb Z$, ...
4
votes
2answers
116 views

Find the number of polynomial zeros of $z^4-7z^3-2z^2+z-3=0$.

Find the number of solutions of $$z^4-7z^3-2z^2+z-3=0$$ inside the unit disc. The Rouche theorem fails obviously. Is there any other method that can help? I have known the answer by Matlab, but ...
0
votes
0answers
39 views

root of equation using bisection method

We have the equation $f(x)=x^3-7x^2 + 14x -6 $ .I have to find the root of the equation using bisection method in the interval of $]1.3 , 2 [$ First I find $f(1.3)=2.567 >0 $ and $f(2)=2>0$ I ...
3
votes
1answer
85 views

What is the value of this integral (using the Argument Principle),

F(z) is given by $$F(z) = e^zz^{-2}(z-1)(z^2-4)(z+8)^7$$ What is the value of the integral $$\int_0^{2\pi} \frac{F'(3e^{i\theta})}{F(3e^{i\theta})}d\theta \space \space ?$$ I think the relevant ...
5
votes
6answers
172 views

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$?

How $\frac{1}{\sqrt{2}}$ can be equal to $\frac{\sqrt{2}}{2}$? I got answer $\frac{1}{\sqrt{2}}$, but the real answer is $\frac{\sqrt{2}}{2}$. Anyway, calculator for both answers return same numbers. ...
0
votes
2answers
90 views

How to simplify this expression with radicals? $3\sqrt2 - \sqrt{32} + \sqrt{\frac{80}{16}}$

I don't understand how I could calculate this: $3\sqrt2 - \sqrt{32} + \sqrt{\dfrac{80}{16}}$ My answer is $-\sqrt2 + \sqrt5$, but the real answer should be $\dfrac{9-4\sqrt2}{4}$.
1
vote
2answers
260 views

Find coefficient in quartic given product of roots

The product of two of the roots of $$x^4 -11x^3 + kx^2 + 269x - 2001=0$$ is $-69$. Find k. This is a question I have recently received, and I am required to take a test on related questions ...
0
votes
1answer
60 views

Sum of roots: Vietas formulas

The equation $x^4-x^3-1=0$ has roots $\alpha, \beta, \gamma, \delta$. Find the equations with roots $\alpha^6, \beta^6, \gamma^6, \delta^6$. I was able to do this using the substitution $y=x^3$. I ...
1
vote
1answer
49 views

roots of functions involving several sines

Is it possible to find exact solutions (in $\mathbb{R}$) of equations of the type $$\alpha_1\sin(\beta_1 t)+\alpha_2\sin(\beta_2 t)+1=0$$ for $\alpha_i,\beta_i\in\mathbb{R}$? In a comment to this ...
9
votes
1answer
100 views

Replacing numbers by roots of quadratic

We have $10$ numbers in the interval $(0,1)$, not necessarily distinct. At any moment, we can choose two of them, $a$ and $b$. If the quadratic $x^2-ax+b$ has two (possibly identical) real roots, we ...
2
votes
5answers
376 views

Sum of roots: Vieta's Formula

The roots of the equation $x^4-5x^2+2x-1=0$ are $\alpha, \beta, \gamma, \delta$. Let $S_n=\alpha^n +\beta^n+\gamma^n+\delta^n$ Show that $S_{n+4}-5S_{n+2}+2S_{n+1}-S_{n}=0$ I have no idea how to ...
15
votes
3answers
294 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
1
vote
1answer
42 views

Prove/disprove number of zeros inequality

Having a continuous differentiable function $f(x)$, and denote $Z(\cdot)$ number of zeros (assume real line), and $(\cdot)^\prime$ first derivative, I would like to know if following inequality ...
0
votes
2answers
82 views

How to find all solutions of the equation $\sin x+\cos x=0$ which belong to $(-\pi, \pi)$?

Could you please help me understand and answer this question? Find all  the  solutions of this equation $$ \sin x+\cos x=0 $$ which belong  to  the interval $(-π; π)$ Progress Divided by ...
1
vote
0answers
31 views

finding root of 3rd degree math equation

I need to solve the following equation and give a simple formula for $y$ such that with the known value of $x$ we can easily compute value of $y$. $$x = \frac{(c+ky)y^{2}}{2}$$ $c$ and $k$ are ...
9
votes
1answer
127 views

Prove that $ ax^2+bx+c=0 $ has at least one root in $(0,1)$ if $10a+12b+15c=0$

If $10a+12b+15c=0$, Prove that $$ ax^2+bx+c=0 $$ has at least one root in $(0,1)$. Progress I tried to solve this by Rolle`s theorem ($f'$ has a root between any two roots of $f$), but could not ...
1
vote
0answers
63 views

Real roots of an nth order polynomial

Given an nth order polynomial, is there any algorithm that can calculate all the roots ? Is there any algorithm that can calculate ALL the roots of the equation ? ...
1
vote
1answer
146 views

Determining existence of roots of a polynomial in the unit disk (possibly with Rouché's theorem?)

I'm studying for my PhD prelim exam in complex analysis, and I ran into this example problem. Show that the polynomial $$p(z)=z^{47} − z^{23} + 2z^{11} − z^5 + 4z^2 + 1$$ has at least one root ...
5
votes
3answers
112 views

Rules for whether an $n$ degree polynomial is an $n$ degree power

Given an $n$ degree equation in 2 variables ($n$ is a natural number) $$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$ If all values of $a$ are given rational numbers, are there any known ...
2
votes
1answer
32 views

Finding the value of $y=b^2(3a^2+4ab+2b^2)$ if $a^2(2a^2+4ab+3b^2)=3$ and $a$ and $b$ are distinct zeros of $x^3-2x+c$

If $a$ and $b$ are distinct zeroes of the polynomial $x^3-2x+c$ and $$a^2(2a^2+4ab+3b^2)=3$$ $$b^2(3a^2+4ab+2b^2)=y$$ Evaluate $y$ I tried for many hours but couldn't solve this question. ...
12
votes
2answers
192 views

Solve $x^7-5x^4-x^3+4x+1=0$ for $x$

Solve for $x$ $$x^7-5x^4-x^3+4x+1=0$$ This equation has been bugging me since the past few days. I have found, using the Rational Root Theorem that $x=1$ is a root of this equation. However, ...
16
votes
1answer
382 views

Something strange about $\sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ and its friends

We have the nice radical identity involving $d = 163$, $$-\sqrt{ 44- \sqrt{ 44 - \sqrt{ 44-x}}}=x,\quad\quad x = 2-2\sum_{n=1}^{27}\cos\left(\frac{2\pi\, t_1(n)}{163}\right)=-6.15824\dots$$ where ...